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'(1) Find \\(\\lim _{h \\rightarrow 0} \\frac{f(a+2 h)-f(a-h)}{h}\\).\n(2) Let x-a=h, then x=a+h, as x \\longrightarrow a, h \\longrightarrow 0. Find the following expression:\n\\[\egin{aligned}\\lim _{x \\rightarrow a} \\frac{x^{2} f(a)-a^{2} f(x)}{x^{2}-a^{2}}\\end{aligned}\\]'

'Find the following limits.'

'Find the following limits. (1) $\\lim _{x \\rightarrow 1} \\frac{x^{2}-1}{x-1}$ (2) $\\lim _{x \\rightarrow 2} \\frac{3 x^{2}-4 x-4}{2 x^{2}-7 x+6}$ (3) $\\lim _{x \\rightarrow 0} \\frac{\\sqrt{x+4}-2}{x}$'

'Find the following limits. Note that \ a \ in (3) is a constant.'

'Find the following limits, where (a) is a constant.'

'Explain the methods of proving equations and inequalities.'

'Derivative and its calculation: Explain the definition for finding the derivative.'

'(20) Determine sequence coefficients from limit conditions\nDetermine the coefficients of the sequence from the limit conditions.\nExample: In order for the sequence {an} to converge, a certain coefficient a needs to be predetermined. Find this coefficient.'

'Find the following limit.'

'Famous function and the associated limit'

'Find the following limit.'

'Find the following limits.'

'Practice: Let the curve y=√(4-x) be denoted as C. For t (2≤t≤3), consider the points (t,√(4-t)) on curve C, the origin, and the point (t,0) to form a triangle with an area denoted as S(t). Divide the interval [2,3] into n equal parts, where the endpoints and division points are represented in ascending order as t₀=2, t₁, t₂, ⋯, tₙ₋₁, tₙ=3, then find the limit value of limₙ→∞(1/n ∑ₖ=1ⁿ S(tₖ)).'

"Let's learn about definite integrals and limits of sums, and inequalities."

'Find the following limits: (1) $\\lim _{x \\rightarrow \\frac{\\pi}{2}} \\frac{\\cos x}{2 x-\\pi}$ (2) $\\lim _{x \\rightarrow \\infty} x \\sin \\frac{1}{x}$ (3) $\\lim _{x \\rightarrow 0} x^{2} \\sin \\frac{1}{x}$'

'Define a sequence {In} by the relation I0 = ∫₀¹ e^(-x) dx, In = (1/n!) ∫₀¹ x^n e^(-x) dx (n=1,2,3,......). Answer the following questions:\n (1) Find I0 and I1.\n (2) Express In-In-1 in terms of n for n≥2.\n (3) Find the limit lim(n→∞) In.\n (4) Define Sn=∑(k=0,n) 1/k!. Find lim(n→∞) Sn.'

'When the sequence {an}(n=1,2,3,⋯⋯) satisfies lim_{n→∞}((3n-1)an)=-6, then lim_{n→∞}nan= \\ square.'

'Find the limit of the function: Find the function f(x) under the following conditions:\n1. \\( \\lim _{x \\rightarrow \\infty} \\frac{f(x)-2 x^{3}+3}{x^{2}}=4 \\).\n2. \\( \\lim _{x \\rightarrow 0} \\frac{f(x)-5}{x}=3 \\).'

'Find the limit of the sequence defined by the nth term with the following expressions:'

'Find the following limits:\n(1) \\(\\lim_{n \\to \\infty}\\frac{3+7+11+\\cdots+(4n-1)}{3+5+7+\\cdots+(2n+1)}\\)\n(2) \\(\\lim_{n \\to \\infty}\\left\\{\\log_{3}\\left(1^{2}+2^{2}+\\cdots+n^{2}\\right)-\\log_{3}n^{3}\\right\\}\\)\n(2) Tokyo Denki University'

'Find the following limit.'

'Consider the sequence {an(x)}, where an(x)=sin^{2n+1} x/sin^{2n} x+cos^{2n} x (0≤x≤π).'

'Find the limit of the sequences (2)...irrational expressions, etc.'

'Find the following limits: (1) $\\lim_{x \\rightarrow \\infty}\\left\\{\\frac{1}{2} \\log_{3} x+\\log_{3}(\\sqrt{3 x+1}-\\sqrt{3 x-1})\\right\\}$ (2) $\\lim_{x \\rightarrow -\\infty}(\\sqrt{x^{2}+3 x}+x)$'

'The limit of the function \ y=x e^{-x} \ is \ \\lim _{x \\rightarrow \\infty} x e^{-x}=0 \.'

'Find the limit related to area.'

'Find the limit of the sequence {an}.'

'Let { an(x)} be a sequence defined by an(x)=sin ^{2 n+1} x / (sin ^{2 n} x +cos ^{2 n} x) (0 ≤ x ≤π).'

'Find the limit of \\( \\lim _{n \\rightarrow \\infty} \\frac{\\log \\left(1^{1} \\cdot 2^{2} \\cdot 3^{3} \\cdots \\cdots \\cdot n^{n}\\right)}{n^{2} \\log n} \\).'

'Explain the continuity and differentiability of functions.'

'Find the following limits.'

'Find the limits of the sequence {an}'

'If the differential coefficient exists, \\( f(x) \\) is differentiable at \ x=a \'

'Investigate the limit of the sequence 1/2, 2/3, 3/4, 4/5, ...'

'Find the limits of the following sequence:\n\nFor the sequence { n^k }, find the limits when k is a positive integer, a positive rational number, and a positive irrational number, respectively.'

'Find the limit of the sequence represented by the nth term.'

'Find the limit of the sequence represented by the following expressions for the nth term.'

'Find the following limits.'

'Find the following limit.'

'Find the limit of \ \\lim _{n \\rightarrow \\infty} \\frac{\\cos n \\pi}{n} \.'

'Find the limit of the sequence represented by the nth term with the following expressions.'

'Please solve the problem of finding the limit of: \\( \\lim_{{x \\to a}} f(x) = \\alpha \\)'

'For a real number x, let [x] denote the integer m satisfying m ≤ x < m+1. Find the limit as n approaches infinity of [10^{2n}π] / 10^{2n}. [University of Yamanashi]'

'Find the limit of the sequence represented by the following expressions for the nth term.'

'Find the following limit.'

'For a real number x, let [x] denote the largest integer that does not exceed x. Let n be a positive integer, and define an=∑(k=1 to n) [√(2n^2-k^2)]/(n^2). Find lim(n approaching infinity) an.'

'Find the following limits.'

'Find the limit of the following sequences:'

'Find the following limits.'

'(2) Let $S_n = \\frac{1}{\\sqrt{n}} \\sum_{k=n+1}^{2n} \\frac{1}{\\sqrt{k}}$, then $S_n = \\sum_{k=n+1}^{2n} \\frac{1}{\\sqrt{\\frac{k}{n}}} \\cdot \\frac{1}{n}$. Since $S_n$ represents the sum of the areas of rectangles, it follows that $S = \\lim_{n \\to \\infty} S_n = \\int_{1}^{2} \\frac{dx}{\\sqrt{x}} = [2 \\sqrt{x}]_{1}^{2} = 2(\\sqrt{2}-1)$. An alternative approach is $S = \\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{k=1}^{n} \\frac{1}{\\sqrt{1+\\frac{k}{n}}} = \\int_{0}^{1} \\frac{dx}{\\sqrt{1+x}} = [2 \\sqrt{1+x}]_{0}^{1} = 2(\\sqrt{2}-1)$.'

'(2) Let {an} be a sequence of positive integers with n digits. Find the limit lim(n→∞) (log10an)/n. [Hiroshima City University]'

'Please verify the convergence of the given sequence {n^k} (k>0).'

'Limit from one side of a function'

'In what ways can red charts help solidify mathematical skills?'

'Find the limit of 62.'

'64\n\\[\n\\text { (1) } \egin{array}{ll}\nf^{\\prime}(x)=\\lim _{h \\rightarrow 0} \\frac{f(x+h)-f(x)}{h} \\\\\n= & \\lim _{h \\rightarrow 0} \\frac{\\{2(x+h)-3\\}-(2 x-3)}{h} \\\\\n=\\lim _{h \\rightarrow 0} \\frac{2 h}{h}=\\lim _{h \\rightarrow 0} 2=2\n\\end{array}\n\\]'

"Derivatives and Derivative Functions\nDerivatives\nD Average rate of change ( f(b)-f(a) / b-a )(a ≠ b)\nD Derivative (Rate of change)\nf'(a)=lim(b → a) (f(b)-f(a))/(b-a)=lim(h → 0) (f(a+h)-f(a))/h"

'Translate the given text into multiple languages.'

'(4) It is permissible to use $\\lim _{x \\rightarrow+0} x \\log x(\\lim _{x \\rightarrow \\infty} \\frac{\\log x}{x}=0)$.'

'Explain the definition of continuity of a function.'

'By repeating, we have $a_{n}=1-\\left(\\frac{1}{2}\\right)^{2 n-1}$. Therefore, $\\lim_{n \\rightarrow \\infty} a_{n}=1$.'

'Find the limit of functions represented by definite integrals'

''

'When the sequences {a_n} and {b_n} converge, the following hold:'

'Find the following limits. Where a is a constant.'

'Find the limits of the following sequences: (1) {2^{n} / n} (2) {n^{2} / 3^{n}}'

'Let the asymptote be represented by the straight line y=ax+b, then the limit of f(x)/x as x approaches positive or negative infinity is a, and the limit of f(x)-ax is b.'

'Find the following limits.'

'Prove the following properties for converging sequences {an},{bn} where lim(n→∞)an=α and lim(n→∞)bn=β:\n1. Constant multiple lim(n→∞)k an=kα\n2. Sum - Difference lim(n→∞)(an+bn)=α+β, lim(n→∞)(an-bn)=α-β'

'Prove the following equation. \\[ \\lim_{b \\to a} \\frac{c-a}{b-a} = \\lim_{b \\to a} \\frac{b+2a}{\\sqrt{3}(\\sqrt{a^2+ab+b^2} + \\sqrt{3}a)} = \\frac{1}{2} \\]'

'Prove that the limit of the sequence {r^{n} / n^{k}},{n^{k} / r^{n} } as r>1, lim _{n へ ∞} r^{n} / n^{2}=∞。'

'Mathematics III\n251\n\\\lim _{\\frac{\\pi}{n} \\rightarrow 0} \\frac{\\sin \\frac{\\pi}{n}}{\\frac{\\pi}{n}}=1, \\quad \\lim _{\\frac{\\pi}{n} \\rightarrow 0} \\frac{1}{\\cos \\frac{\\pi}{n}}=1\\n\nThus, as \ n \\longrightarrow \\infty \, the convergence to a non-zero value for \\( n^{k}\\left(b_{n}-a_{n}\\right) \\) occurs when \ k-2=0 \ i.e., when \ k=2 \ and \\( \\lim _{n \\rightarrow \\infty} n^{2}\\left(b_{n}-a_{n}\\right)=\\pi \\)'

'Mathematics III'

'Example 329 | One-sided limits and existence of limits'

'\n(2)\n\\[\n\egin{array}{l}\n\\lim _{h \\rightarrow+0} \\frac{f(0+h)-f(0)}{h}=\\lim _{h \\rightarrow+0} \\frac{\\sin h-0}{h}=\\lim _{h \\rightarrow+0} \\frac{\\sin h}{h}=1 \\\\\n\\lim _{h \\rightarrow-0} \\frac{f(0+h)-f(0)}{h}=\\lim _{h \\rightarrow-0} \\frac{\\left(h^{2}+h\\right)-0}{h}=\\lim _{h \\rightarrow-0}(h+1)=1\n\\end{array}\n\\]\n\h \\longrightarrow+0\ and \h \\longrightarrow-0\ have the same limit value, and \\(f^{\\prime}(0)=1\\) which means that \\(f(x)\\) is differentiable at \x=0\.\nTherefore, \\(f(x)\\) is continuous at \x=0\.'

"Using L'Hopital's Rule, find the following limits."

'When the sequences {a_{n}}, {b_{n}} converge, the following holds:'

'Find the following limits.'

'Since $\\lim _{x \\rightarrow \\infty} y=\\lim _{x \\rightarrow \\infty}\\left(\\sqrt{x^{2}+1}-x\\right)=\\lim _{x \\rightarrow \\infty} \\frac{1}{\\sqrt{x^{2}+1}+x}=0$, the line $y=0$ is an asymptote.'

'Example 19 | Recursive Formula and Limit (3)'

'The n-th term a_{n} is a_{n} = \\frac{3n-2}{n+1}, Therefore, \\lim _{n \\rightarrow \\infty} a_{n} = \\lim _{n \\rightarrow \\infty} \\frac{3n-2}{n+1} = \\lim _{n \\rightarrow \\infty} \\frac{3-\\frac{2}{n}}{1+\\frac{1}{n}} = 3 \\neq 0, Hence, this infinite series diverges.'

'Consider the sequence {an}, where the nth term an is an n-digit positive integer. Find the limit lim (n→∞) (log10 an)/n.'

'Find lim_{n → ∞} Σ_{k=1}^{2n} (1 + k/n)^p * 1/n and lim_{n → ∞} Σ_{k=1}^{2n} (k/n)^p * 1/n.'

'Find the following limits.'

'Find the limit of the sequence {r^n}.'

'Prove that if a curve has at most 1 inflection point, then the curve does not have a cusp.'

'Find the following limit. (a) lim_{x \\rightarrow -\\infty} \\frac{4^x}{3^x - 2^x}'

'Comprehensive Exercise'

'Prove that when the term number n of the infinite sequence {an} approaches infinity, if a_n approaches a constant value α, then lim{n -> ∞} a_n=α, or as n approaches infinity, a_n approaches α, and denote α as the limit value of the sequence {an}. Prove this statement.'

'Find the limits of the sequences {r^n / n^k}, {n^k / r^n}.'

'If a function f(x) is continuous for all values of x in its domain, how is it expressed?'

'When x > 1, the inequality 0 < log x < x holds. Using this inequality, find the limit lim _{x \\rightarrow ∞} \\\frac{\\log x}{x}\. Here, log x is the natural logarithm with base e = 2.71828....'

'Find the following limits.'

''

'Find the following limit.'

'Investigate whether the following functions are continuous and differentiable at x = 0:'

'Find the limit of (3) lim _{x \\rightarrow 0} x^{2} \\sin \\frac{1}{x}'

"Exercise Problem 22 Prove Wallis's Formula, Sterling's Formula"

'(1) Calculate $\\lim_{x \\to -\\infty} \\frac{3^{x}+5^{x}}{3^{x}-5^{x}}$'

'For a real number x, let [x] be the integer m that satisfies m ≤ x < m+1. Find the limit as n approaches infinity of [10^(2n)π] / 10^(2n).'

'Find the limits of the following sequences. (A) \ -2 n^{2}+3 n+1 \ (B) \ \\frac{-5 n+3}{3 n^{2}-1} \ (C) \ \\frac{2 n^{2}-3 n}{4 n^{2}+2} \'

'Explain about the limit from one side of a function.'

'Find the following limit.'

'Find the following limits.'

'Find the following limits. Where α is a constant.'

'Find the given limit.'

'(1) Find the values of the constants \ a, b \ that satisfy the equation \ \\lim _{x \\rightarrow 3} \\frac{a x^{2}+b x+3}{x^{2}-2 x-3}=\\frac{5}{4} \.\n(2) Express \\( \\lim _{h \\rightarrow 0} \\frac{f(a+2 h)-f(a-h)}{h} \\) in terms of \\( f^{\\prime}(a) \\).'

'(1) Find the values of the constants a, b that satisfy the equation \ \\lim _{x \\rightarrow 1} \\frac{x^{2}+a x+b}{x-1}=3 \. (2) Express \\( \\lim _{h \\rightarrow 0} \\frac{f(a-3 h)-f(a)}{h} \\) using \\( f^{\\prime}(a) \\).'

'Find the following limits.'

'Find the following limits.'

'Find the following limits.'

'Find the following limits.'

'Practice finding the following limits.'

'Find the following limits.'

'Prove the inequality of the sum of series and find the limit (1) For natural numbers n greater than or equal to 2, prove the following inequality.'

'For the general term an and the sum Sn from the first term to the nth term:'

'Find the following limit.'

'(4) The limit from the right is 0, from the left is 1; the limit does not exist'

'For a real number x, let [x] denote the integer m that satisfies m≤x<m+1. Find the value of lim n→∞ [10^2nπ]/10^2n as n approaches infinity.'

'(3) (A) \ \\lim _{x \\rightarrow \\infty} \\frac{2^{x}a-2^{-x}}{2^{x+1}-2^{-x-1}} =\\lim _{x \\rightarrow \\infty} \\frac{a-\\frac{1}{2^{2 x}}}{2-\\frac{1}{2^{2 x+1}}} =\\frac{a}{2} \ Therefore, \ \\quad \\frac{a}{2} =\\frac{3}{4} \ hence \ \\quad a=\\frac{3}{2} \'

'Find \\( f(x) \\) when the function \\( f(x) \\) satisfies \\( \\lim _{x \\rightarrow \\infty} \\frac{f(x)-2 x^{3}+3}{x^{2}}=4, \\lim _{x \\rightarrow 0} \\frac{f(x)-5}{x}=3 \\).'

'Find the following limits. Where a is a constant.'

'Find the value of \\( \\lim _{n \\rightarrow \\infty} \\frac{\\log \\left(1^{1} \\cdot 2^{2} \\cdot 3^{3} \\cdots \\cdots \\cdot n^{n}\\right)}{n^{2} \\log n} \\).'

'Find the following limit. $\\ lim _ {n \\ to \\ infty} \\ frac {1} {n ^ {2}} \\ left(e ^ { \\ frac {1} {n}} + 2 e ^ { \\ frac {2} {n}} + 3 e ^ { \\ frac {3} {n}} + \\ cdots \\ cdots + n e ^ { \\ frac {n} {n}} \\ right)$'

'Find the limit of \ S_{n} \ as \ n \ approaches infinity.'

'Find the maximum and minimum values of the following functions. If necessary, you can use \ \\lim _{x \\rightarrow \\infty} x e^{-x}=\\lim _{x \\rightarrow \\infty} x^{2} e^{-x}=0 \ in (2).'

'Find the following limits.'

'Find the following limit.'

'Therefore, $\\quad a_{n+1}-a_{n}=2\\left(\\frac{1}{4}\\right)^{n-1}$, $\\quad a_{n+1}-\\frac{1}{4} a_{n}=\\frac{11}{4}$ Subtracting the two equations gives $a_{n}=\\frac{11}{3}-\\frac{8}{3}\\left(\\frac{1}{4}\\right)^{n-1}$, eliminating $a_{n+1}$. So, $\\lim _{n \\rightarrow \\infty} a_{n}=\\lim _{n \\rightarrow \\infty}\\left\\{\\frac{11}{3}-\\frac{8}{3}\\left(\\frac{1}{4}\\right)^{n-1}\\right\\}=\\frac{11}{3}-0=\\frac{11}{3}$'

'Prove that the function f(x) with the given conditions is not differentiable at x=π/2. When 112 x ≤ π/2, f(x)=cos x-π/2 sin x and when x>π/2, f(x)=x-π'

'Find the following limits.'

'Find the following limits. (2) where \ p>0 \.\n(1) \\( \\lim _{n \\rightarrow \\infty} \\frac{1}{n}\\left\\{\\left(\\frac{1}{n}\\right)^{2}+\\left(\\frac{2}{n}\\right)^{2}+\\left(\\frac{3}{n}\\right)^{2}+\\cdots \\cdots+\\left(\\frac{3 n}{n}\\right)^{2}\\right\\} \\)\n(2) \\( \\lim _{n \\rightarrow \\infty} \\frac{(n+1)^{p}+(n+2)^{p}+\\cdots \\cdots+(n+2 n)^{p}}{1^{p}+2^{p}+\\cdots \\cdots+(2 n)^{p}} \\)'

'Find the limit.'

'Find the limit of the sequence represented by the following formula for the n-th term. \n\n(A) \ \\sqrt{4n-2} \ \n(B) \ \\frac{n}{1-n^{2}} \ \n(C) \\( n^{4}+(-n)^{3} \\) \n(D) \ \\frac{3n^{2}+n+1}{n+1}-3n \'

'Determine the value of the constant a such that the equation $\\lim _{x \\rightarrow 0} \\frac{\\sqrt{x^{2}+1}-(a x+1)}{x}=3$ holds.'

'Find the following limits.'

'Find the following limit (1) \ \\lim _{n \\rightarrow \\infty} \\sum_{k=1}^{n} \\frac{\\pi}{n} \\sin ^{2} \\frac{k \\pi}{n} \'

'Solve the following problem:'

'(1) Let \\( \\left\\{a_{n}\\right\\}(n=1,2,3, \\cdots \\cdots) \\) be a sequence such that \\( \\lim _{n \\rightarrow \\infty}(3 n-1) a_{n}=-6 \\), then\n\ \\lim _{n \\rightarrow \\infty} n a_{n}=\\square \\text{ is } \'

'(1) Calculate $\\lim _{n \\rightarrow \\infty} \\frac{1}{n}\\left\\{\\left(\\frac{1}{n}\\right)^{2}+\\left(\\frac{2}{n}\\right)^{2}+\\left(\\frac{3}{n}\\right)^{2}+\\cdots \\cdots+\\left(\\frac{3 n}{n}\\right)^{2}\\right\\}$'

'Find the following limit. \ [x] \ denotes the greatest integer that is less than or equal to \ x \. (1) \\( \\lim _{x \\rightarrow k-0}([2x]-2[x]) \\) (where \ k \ is an integer).'

'Explain the relationship between differentiability and continuity.'

'Investigate whether the function is continuous and differentiable at x=0. (2) f(x)=\\left\\{\egin{array}{ll}0 & (x=0) \\\\ \\frac{x}{1+2^{\\frac{1}{x}}} & (x \\neq 0)\\end{array}\\right\\}'

'Determine the values of constants a and b such that the equation holds. \\\lim _{x \\rightarrow 1} \\frac{a \\sqrt{x+1}-b}{x-1}=\\sqrt{2}\'

'Determine the values of the constants a and b so that the following equations hold true.'

'Explain the concepts of convergence and divergence in limits, and describe the basic properties of how sequences behave.'

'Find the following limits.'

'Find the function f(x) when it satisfies lim_{x \\rightarrow \\infty} \\frac{f(x)-2 x^{3}+3}{x^{2}}=4, lim_{x \\rightarrow 0} \\frac{f(x)-5}{x}=3.'

'Limit of a function'

'Limits of trigonometric functions\nWhen the unit of angle is in radians \ \\lim _{x \\rightarrow 0} \\frac{\\sin x}{x}=1, \\lim _{x \\rightarrow 0} \\frac{x}{\\sin x}=1 \'

'(1) The limit exists from both the right side and the left side;'

'Find the limit of the sequence represented by the nth term.'

'Explain the meaning of x approaching a+0 and x approaching a-0 in the function f(x), and whether the limit of the function exists when these are the same or different.'

'Explain the relationship between the sequences {a_n} and {b_n} when they converge and their limits as n approaches infinity are a_n = α, b_n = β.'

"Using L'Hopital's rule, find the following limits."

'Find the following limit. (4): $\\lim_{x \\to \\infty} (\\sqrt{x^{2} + 2x} - x)$'

'Evaluate the following limit.'

'Let \\( \\left\\{a_{n}(x)\\right\\} \\) be the sequence defined by \\( a_{n}(x)=\\frac{\\sin ^{2 n+1} x}{\\sin ^{2 n} x+\\cos ^{2 n} x}(0 \\leqq x \\leqq \\pi) \\). (1) Find the limit \\( \\lim _{n \\rightarrow \\infty} a_{n}(x) \\) of this sequence. (2) Let \\( \\lim _{n \\rightarrow \\infty} a_{n}(x) \\) be denoted by \\( A(x) \\), plot the graph of the function \\( y=A(x) \\).'

'For a continuous function on a closed interval, the intermediate value theorem holds. In other words, for a continuous function f(x) on a closed interval [a, b], for any value k between f(a) and f(b), there exists a c such that f(c) = k. When this condition is not met, consider the function f(x) = sin(1/x) assumed to be continuous on the interval (0, 1], and explain the scenario where there is no c such that f(c) = k for a certain k.'

'If S is the limit, then find S.\nS = \\lim_{n\\rightarrow \\infty} \\frac{1}{n^{2}}\\left\\{\\sqrt{(2n)^{2}-1^{2}}+\\sqrt{(2n)^{2}-2^{2}}+\\sqrt{(2n)^{2}-3^{2}}+\\cdots \\cdots+\\sqrt{(2n)^{2}-(2n-1)^{2}}\\right\\}'

'Assuming that every year in the future, one third of the people living outside Tokyo move into the city, and one third of the people living in the city move out. Let the population of people outside the city in the nth year be an and inside the city be bn. Find lim n→∞ an/bn. It is assumed that the total population of both inside and outside the city remains constant regardless of the year.'

'Define S as the following limit, find the value of S.'

'102 a=4, b=-1, limit -\\frac{15}{8}'

'Given the following conditions, find the coordinates and velocity trajectory of point Q. When point P moves along the x-axis from the origin (0,0) to (π,0) at a speed of π per second, find the velocity v(t) of point Q after t seconds.'

'Find the following limits.'

'Chapter 4 Limits Exercises'

'Continuity of a function'

'Prove that for an infinite sequence {an} (n=1,2,⋯⋯), the limit as n approaches infinity of 1/n^k is 0.'

'Investigate the limit of the sequence \ \\left\\{a_{n}\\right\\} \ given by \ \\lim _{n \\rightarrow \\infty} \\frac{a_{2}+a_{4}+\\cdots \\cdots+a_{2 n}}{a_{1}+a_{2}+\\cdots \\cdots+a_{n}} \.'

'Find the following limit.'

'Mathematics \nClearing the denominator, we have (c^{2}-1)(x+1)=c^{2}(x-1) thus 2 c^{2}=x+1 hence c^{2}=\\frac{x+1}{2} \nx>1, c>1 therefore c=\\sqrt{\\frac{x+1}{2}} \n\\lim _{x \\rightarrow 1+0} \\frac{c-1}{x-1}=\\lim _{x \\rightarrow 1+0} \\frac{\\sqrt{\\frac{x+1}{2}}-1}{x-1}=\\lim _{x \\rightarrow 1+0} \\frac{\\frac{x+1}{2}-1}{(x-1)\\left(\\sqrt{\\frac{x+1}{2}}+1\\right)} =\\lim _{x \\rightarrow 1+0} \\frac{1}{2\\left(\\sqrt{\\frac{x+1}{2}}+1\\right)}=\\frac{1}{2(\\sqrt{1}+1)}=\\frac{1}{4} \n\\lim _{x \\rightarrow \\infty} \\frac{c-1}{x-1}=\\lim _{x \\rightarrow \\infty} \\frac{\\sqrt{\\frac{x+1}{2}}-1}{x-1}=\\lim _{x \\rightarrow \\infty} \\frac{\\sqrt{\\frac{1}{2}\\left(1+\\frac{1}{x}\\right)}-\\frac{1}{\\sqrt{x}}}{\\sqrt{x}-\\frac{1}{\\sqrt{x}}}=0'

'Find the limit of the sequence represented by the n-th term.'

'To make the limit \\(\\lim_{x \\rightarrow 0} \\frac{\\sqrt{9-8 x+7 \\cos 2 x}-(a+b x)}{x^{2}}\\) have a finite value, determine the values of the constants \a, \\quad b\ and find the limit value.'

'Find the following limit.'

'(2) Find the value of the constant \ a \ when \\( \\lim _{n \\rightarrow \\infty}\\left(\\sqrt{n^{2}+a n+2}-\\sqrt{n^{2}-n}\\right)=5 \\).'

'Find the limit of the sequence represented by the nth term.'

'Comprehensive Exercise'

'Find \\( \\lim _{x \\rightarrow 0} \\frac{1}{x^{3}}\\left\\{\\sqrt{1+2 x}-\\left(1+x-\\frac{x^{2}}{2}\\right)\\right\\} \\).'

'Find the following limits.\n\n(1) lim_{n→∞} \\frac{1}{n^{2}} \\left\\{ \\sqrt{(2 n)^{2}-1^{2}}+\\sqrt{(2 n)^{2}-2^{2}}+\\cdots \\cdots+\\sqrt{(2 n)^{2}-(2 n-1)^{2}} \\right\\} \n(2) lim_{n→∞} sum_{k=1}^{2 n} \\frac{n}{2 n^{2}+3 n k+k^{2}}\n\n〔(1) Yamaguchi University, (2) Shibaura Institute of Technology〕'

'Let the sequence \\( \\left\\{a_{n}(x)\\right\\} \\) be defined as \\( a_{n}(x)=\\frac{\\sin ^{2 n+1} x}{\\sin ^{2 n} x+\\cos ^{2 n} x}(0 \\leqq x \\leqq \\pi) \\).\n(1) Find the limit of this sequence, \\( \\lim _{n \\rightarrow \\infty} a_{n}(x) \\).\n(2) Let \\( \\lim _{n \\rightarrow \\infty} a_{n}(x) \\) be denoted as \\( A(x) \\). Plot the graph of the function \\( y=A(x) \\).\n〔Meijo University〕'

'Find the value of the constant \ a \ such that the equation \\( \\lim _{x \\rightarrow 0} \\frac{\\sqrt{x^{2}+1}-(a x+1)}{x}=3 \\) holds.'

'Investigate the limit of the sequence 1/2, 2/3, 3/4, 4/5.'

'Find the following limits.'

'Find the following limit.'

'Find the following limits. (1) $\\lim _{x \\rightarrow 0} \\frac{\\sqrt{\\cos 5 x}-\\sqrt{\\cos 3 x}}{x^{2}}$(2) $\\lim _{x \\rightarrow \\frac{1}{4}} \\frac{\\tan \\pi x-1}{4 x-1}$'

'Limit of a sequence'

'Find the limit \\lim _{n \\rightarrow \\infty} T_{n}.'

"The following propositions regarding limits may seem ambiguous, but in reality, they are all false. Let's find out when they do not hold by examining counterexamples."

'(1) As \ x \\rightarrow \\infty \, \\( \\{ \\log _{\\frac{3}{2}}(2 x)-\\log _{\\frac{3}{2}}(3 x+2) \\} =\\lim _{x \\rightarrow \\infty} \\log _{\\frac{3}{2}} \\frac{2 x}{3 x+2} \\) = \\( \\lim _{x \\rightarrow \\infty} \\log \\frac{3}{2} \\frac{2}{3+\\frac{2}{x}}=\\log _{\\frac{3}{2}} \\frac{2}{3}=\\log _{\\frac{3}{2}}(\\frac{3}{2})^{-1}=-1 \\) \ \\leftarrow \ Divide numerator and denominator by \ 2^{x} \.'

'Find the following limits.'

'(2) \ \\quad( \ And the expression \\()=\\lim _{n \\rightarrow \\infty}\\left\\{\\log _{2} \\frac{1}{4} n^{2}(n+1)^{2}-\\log _{2}\\left(n^{4}+1\\right)\\right\\} \\)'

'Differentiation coefficient'

'(2) The limit from the right is \ \\infty \, from the left is \ -\\infty \; the limit does not exist'

'Find the values of the 3 limits. where $a$ is a constant.\n(1) $\\lim _{x \\rightarrow 0} \\frac{a^{x}-1}{x}(a>1)$\n(2) $\\lim _{x \\rightarrow 0} \\frac{\\log (a+x)-\\log a}{x}(a>0)$\n(3) $\\lim _{x \\rightarrow 0} \\frac{\\log (1+a x)}{x}$\n(4) $\\lim _{x \\rightarrow 0} \\frac{1-\\cos 2 x}{x \\log (1+x)}$'

'Find the limits of the following sequences. (A) 1, \\frac{1}{2^{2}}, \\frac{1}{3^{2}}, \\frac{1}{4^{2}}, (B) \\sqrt{2}, \\sqrt{5}, \\sqrt{8}, \\sqrt{11}, \\cdots \\cdots'

'Practice finding the limit of the following sequences.'

'Find the following limits. Note that a and b are constants. (1) University of Otaru, (2) Tokyo Denki University'

'(2) Calculate \ \\lim _{x \\rightarrow \\infty} \\frac{\\left[ \\sqrt{x+x^{2}} \\right] - \\sqrt{x}}{x} \.'

'Find the limit of $\\lim _{n \\rightarrow \\infty} \\frac{1}{n} \\sin \\frac{n \\pi}{4}$.'

'For the given sequence {a_n}, find the limit \\\lim _{n \\rightarrow \\infty} \\frac{a_{2}+a_{4}+\\cdots \\cdots+a_{2 n}}{a_{1}+a_{2}+\\cdots \\cdots+a_{n}} \。'

'Calculate the following limit value of S.'

'(1) For the sequence {an} that satisfies the following relation, find lim(n→∞)an and lim(n→∞)nan. 30(a): lim(n→∞)(2n-1)an=1'

'Find the following limits.'

'Prove that the function f(x) is not differentiable at x=π/2.'

'Solve the following problem:'

'Find the limit of the equation \\( \\lim _{x \\rightarrow 0} \\frac{1}{x^{3}}\\left\\{\\sqrt{1+2 x}-\\left(1+x-\\frac{x^{2}}{2}\\right)\\right\\} \\).'

'Find the following limits: (a) $\\lim _{n \\rightarrow \\infty} \\frac{\\sin ^{n} \\theta-\\cos ^{n} \\theta}{\\sin ^{n} \\theta+\\cos ^{n} \\theta} \\quad\\left(0<\\theta<\\frac{\\pi}{4}\\right)$ (b) $\\lim _{n \\rightarrow \\infty} \\frac{r^{n-1}-3^{n+1}}{r^{n}+3^{n-1}}$ (where $r$ is a positive constant) (2) Let $0 \\leqq \\theta \\leqq \\pi$. Define $a_{n}=\\left(4 \\sin ^{2} \\theta+2 \\cos \\theta-3\\right)^{n}$. Find the range of values for $\\theta$ where the sequence $\\left\\{a_{n}\\right\\}$ converges.'

'Find the following limits.'

'Find the following limits.'

'Find the limit.'

'(1) Continuous but not differentiable\n(2) Continuous and differentiable'

'\\[f(x)=\\tan (\\pi x) \\text { then } \\lim _{x \\rightarrow \\frac{1}{4}} \\frac{\\tan (\\pi x)-1}{4 x-1}=\\lim _{x-\\frac{1}{4}} \\frac{1}{4} \\cdot \\frac{f(x)-f\\left(\\frac{1}{4}\\right)}{x-\\frac{1}{4}}=\\frac{1}{4} f^{\\prime}\\left(\\frac{1}{4}\\right) f^{\\prime}(x)=\\frac{\\pi}{\\cos ^{2}(\\pi x)} \\text { so } \\quad f^{\\prime}\\left(\\frac{1}{4}\\right)=\\frac{\\pi}{\\cos ^{2} \\frac{\\pi}{4}}=2 \\pi \\text { hence } \\quad \\lim _{x \\rightarrow \\frac{1}{4}} \\frac{\\tan (\\pi x)-1}{4 x-1}=\\frac{1}{4} \\cdot 2 \\pi=\\frac{\\pi}{2}\\]'

'Investigate the limits for x approaching 1-0, x approaching 1+0, and x approaching 1, respectively.'

'Example 16 | Limits of Functions (2)'

'Find the limit. \\( \\lim _{x \\rightarrow-\\infty}(\\sqrt{9 x^{2}+x}+3 x) \\)'

'Example (41) Limits and Derivatives'

'Regarding the limit of the sequence \ \\left\\{n^{k}\\right\\} \, the following holds true.'

'| r | <1, when r approaches infinity lim_{n -> ∞} r^{2 n}=0, lim_{n -> ∞} r^{2 n+1}=0\nTherefore lim_{n -> ∞} (r^{2 n+1}) / (2 + r^{2 n})=0\nWhen r = 1, r^{2 n}=r^{2 n+1}=1\nTherefore lim_{n -> ∞} (r^{2 n+1}) / (2 + r^{2 n}) = 1 / (2 +1) = 1/3\nWhen r =-1, r^{2 n}=(-1)^{2 n}=((-1)^{2})^{n}=1^{n}=1, r^{2 n+1}=r^{2 n}・r=1・(-1)=-1\nTherefore lim_{n -> ∞} (r^{2 n+1}) / (2 + r^{2 n}) = -1 / (2 + 1) = -1/3\n|r| >1, <left| (1/r)<right| <1, hence lim_{n -> ∞} ((1/r)^{2 n}) = 0\nTherefore lim_{n -> ∞} (r^{2 n+1}) / (2 + r^{2 n}) = lim_{n -> ∞} r / (2 ((1/r)^{2 n}) + 1) = r / (2 ・ 0 + 1) = r\nChapter 2 <square>\nPR <left>{(-1)^{n}<right> oscillates, but, <left>{(-1)^{2 n}<right> converges.'

'[2] Example $\\ lim _{n \\rightarrow \\infty} (\\sqrt{n^{2}+2 n-1}-n) = \\lim _{n \\rightarrow \\infty} \\frac{(\\sqrt{n^{2}+2 n-1}-n)(\\sqrt{n^{2}+2 n-1}+n)}{\\sqrt{n^{2}+2 n-1}+n}$'

'Find the limit. Without using the above formula (1). (2) \ \\lim _{x \\rightarrow \\frac{\\pi}{2}} \\frac{\\sin ^{2} x-1}{\\cos x} \'

'Find the following limits:\n\n1. \\\lim _{x \\rightarrow 0} \\frac{x-\\tan x}{x^{3}}\ \n2. \\(\\lim _{x \\rightarrow \\infty} x(1-e^{\\frac{1}{x}})\\) \n3. \\\lim _{x \\rightarrow+0} x \\log x\'

'How to find the limit of a function'

'Find the following limits. [(1) Kyoto University, (2) Tokyo Institute of Technology] (1) \ \\lim _{x \\rightarrow 1} \\frac{\\sqrt[3]{x}-1}{x-1} \ (2) \ \\lim _{x \\rightarrow 0} \\frac{\\sqrt{x^{2}-x+1}-1}{\\sqrt{1+x}-\\sqrt{1-x}} \'

'Find the following limits.'

'Using the definition of \ e \ to compute limits\n\\( \\lim _{h \\rightarrow 0}(1+h)^{\\frac{1}{h}}=e \\), find the following limits:\n(1) \\( \\lim _{x \\rightarrow 0}(1+2 x)^{\\frac{1}{x}} \\)\n(2) \\( \\lim _{x \\rightarrow 0}(1-2 x)^{\\frac{1}{x}} \\)\n(3) \\( \\lim _{x \\rightarrow \\infty}\\left(1+\\frac{4}{x}\\right)^{x} \\)'

'Find the following limits. (1) \ \\lim _{x \\rightarrow 0} \\frac{\\sin 3 x}{x} \ (2) \ \\lim _{x \\rightarrow 0} \\frac{\\tan x^{\\circ}}{x} \ (3) \ \\lim _{x \\rightarrow 0} \\frac{\\sin ^{2} 2 x}{1-\\cos x} \'

'Determine the page number related to the limits of trigonometric functions from the given table.'

'Find the following limit. (2) \ \\lim _{x \\rightarrow 0} \\frac{1-\\cos 2 x}{x \\tan \\frac{x}{2}} \[Osaka Institute of Technology]'

'\ \\lim_{n \\rightarrow \\infty} \\frac{1}{\\sqrt{n}}=0 \ so \\[ \\lim_{n \\rightarrow \\infty} \\frac{(-1)^{n}}{\\sqrt{n}}=0 \\]'

'Answer the following questions.'

'(1) Since the base is \\\sqrt{2}>1\,\n\\[ \\lim _{x \\rightarrow \\infty}(\\sqrt{2})^{x}=\\infty \\]\n(2) Since the base is \0<\\frac{2}{3}<1\,\n\\[ \\lim _{x \\rightarrow \\infty}\\left(\\frac{2}{3}\\right)^{x}=0 \\]'

'Find the limit lim (n→∞) (r^n)/(2+r^(n+1)) when r>-1.'

'Find the following limit:\n\ \n\\lim _{n \\rightarrow \\infty} \\frac{1}{n+1} \\cos \\frac{n \\pi}{3} \n\'

'Example 15 | Limit of a Function (1)'

'Maximum and minimum of a function\nFor a continuous function f(x) on an interval [a, b], the maximum and minimum values are determined by\n[1] The maximum and minimum values of f(x) at a ≤ x ≤ b\n[2] Comparing the values at the ends of the interval, f(a) and f(b)\nNote: To find the maximum and minimum values of f(x) on an interval (a, b), one needs to compare the extreme values of f(x) and the values of lim x→a+0 f(x) and lim x→b-0 f(x). Also, in the case of the interval (a, ∞), a comparison with lim x→∞ f(x) is required.\nIt should be noted that in open intervals, maximum and minimum values may not always exist.'

'Examine whether the following functions are continuous and differentiable at x=0:\n(1) f(x)=√|x|\n(2) f(x)={sin x (x ≥ 0), x^{2}+x (x<0)}'

'Find the limit of the following sequences. 17 (1) {2^n / n} (2) {n^2 / 3^n}'

'Find the following limit. (Shibaura Institute of Technology) \n(1) \\( \\lim _{n \\rightarrow \\infty} \\frac{1}{n \\sqrt{n}}(\\sqrt{2}+\\sqrt{4}+\\cdots \\cdots+\\sqrt{2 n}) \\)'

'Prove the following using the binomial theorem:\n\\(\\lim _{n \\rightarrow \\infty} \\frac{(1+h)^{n}}{n}=\\infty \\)'

'Example 35 (1) Find the limit. \ \\lim _{x \\rightarrow-1} \\frac{x^{2}-x-2}{x^{3}+1} \'

'What are the key points in expression transformation in limits?'

'Using the mean value theorem, find the following limits.'

'Definition of limit of a sequence'

'Solve problems related to the limit of sequences (using inequalities).'

'Chapter 2 Limit 69 Alternative Solution'

'Investigate the limit in the following cases.'

"Using L'Hopital's rule, find the following limit."

'Find the limit of the sequence represented by the nth term.\n(1) n²-n\n(2) (n+1)/(3n²-2)\n(3) 5n²/(-2n²+1)'

'Limit of a sequence'

"Proof of the Mean Value Theorem\n\nThe geometric interpretation of the Mean Value Theorem is that for a graph of a continuous and differentiable function, when two points A and B are taken on the curve, a tangent parallel to the line AB can be drawn at a certain point on the curve between A and B. While this seems intuitively clear from the figure, it is rigorously proven using the following Rolle's Theorem.\n(1) Rolle's Theorem\nIf a function f(x) is continuous on the interval [a, b] and differentiable on the open interval (a, b), and f(a) = f(b), then there exists a real number c such that f'(c) = 0 and a < c < b.\n\nProof [1] Case where f(a) = f(b) = 0\n(A) If f(x) = 0 always on the interval [a, b], then f'(x) = 0 always, and the theorem holds true.\n(1) If there exist values of x where f(x) > 0, then since f(x) is continuous on the interval [a, b], it attains a maximum at a point x = c in this interval. As f(c) > 0 and f(a) = f(b) = 0, c is neither a nor b. Therefore, a < c < b.\nSince f(c) is a maximum, when |Δx| is sufficiently small, f(c + Δx) ≤ f(c), thus Δy = f(c + Δx) - f(c) ≤ 0. Therefore, if Δx > 0, then Δy/Δx ≤ 0, implying as Δx approaches +0, Δy/Δx ≤ 0. If Δx < 0, then Δy/Δx ≥ 0. This gives the result that as Δx approaches +0, -0, or 0, Δy/Δx = 0. Thus, f(x) is constant on the interval, i.e., f'(c) = 0.\n(B) If there are values of x where f(x) < 0 and c is the value of x where f(x) attains the minimum, then similar reasoning to (1) shows that a < c < b and f'(c) = 0.\n[2] In general, if f(a) = f(b), define g(x) = f(x) - f(a). Since f(a) = f(b), it follows that g(a) = g(b) = 0. Similarly, as in case [1], there exists a real number c such that g'(c) = 0 and a < c < b. Since f'(c) = g'(c) = 0, Rolle's Theorem holds true."

'Example 15 | Principle of Scissors (2) (2) Let {a_{n}} be a sequence where the nth term a_{n} is a n-digit positive integer. Find the limit lim _{n → ∞} log _{10} a_{n} / n.'

'Chapter 6 Application of Integration'

'Determine the values of the constants a, b so that the equation holds.'

'Example 17 Show that lim_{n→∞} (r^n / n^2) = ∞ for the sequence {r^n / n^k}, {n^k / r^n} as r > 1.'

'Find the following limits.'

'Explain what is meant by the limit from one side of a function, and symbolically represent the right-hand limit of f(x) as x approaches a from the range x > a.'

'Prove that (1), (2) hold.'

'(1) When the sequence {an} (n=1,2,3,⋯⋯) satisfies lim_{n→∞}(2n-1)an=1, find lim_{n→∞}an and 13lim_{n→∞}nan.\n(2) Find the values of constants a, b when lim_{n→∞}1/(an+b-√{3n^2+2n})=5.'

'Applications of Differential Calculus'

'Find the limit without using the given formula (1).'

'Find the following limit:\n\\(\n\\lim _{n \\rightarrow \\infty} n\\left(\\sqrt{4+\\frac{1}{n}}-2\\right)\n\\)'

'Let f(x)=-log x. For a real number a, find the number of tangent lines of the curve y=f(x) passing through the point (a, 0). You may use lim_{x→+0} x log x=0.'

'Example 35 | Text problem on the limit of trigonometric functions'

'This is a problem of finding the limit of a function. Please determine the limit value of the function f(x) as x approaches a. In particular, consider the right-hand limit \\lim _{x \\rightarrow a+0} f(x) and the left-hand limit \\lim _{x \\rightarrow a-0} f(x).'

'Find the following limits.'

'For constants a, b, find (a, b) such that \\( \\lim _{x \\rightarrow \\infty}\\left\\{\\sqrt{4 x^{2}+5 x+6}-(a x+b)\\right\\}=0 \\) holds.'

'Find \ \\lim _{x \\rightarrow 0} x^{3} \\sin \\frac{1}{x} \.'

'Examine the limit of asymptotes parallel to the y-axis (x=a).'

'Find the following limit:\n\\(\n\\lim _{n \\rightarrow \\infty} \\frac{\\{(n+2)-(n-2)\\}(\\sqrt{n+1}+\\sqrt{n-1})}{\\{(n+1)-(n-1)\\}(\\sqrt{n+2}+\\sqrt{n-2})} \n\\)'

'Determine the values of the constants \a, b\ so that the following equation holds true:\n(1) \ \\lim _{x \\rightarrow 3} \\frac{\\sqrt{4x+a}-b}{x-3}=\\frac{2}{5} \\n[Fukuoka University]'

'Solve problems related to limits of sequences (polynomials and fractions).'

'Find the limit.\\n(2) \ \\lim _{n \\rightarrow \\infty} \\frac{\\pi}{n} \\sum_{k=1}^{n} \\cos \\frac{k \\pi}{2 n} \'

'(1) Find \\( \\lim_{n \\to -2} (-2)\\). \n(2) Find \ \\lim_{n \\to \\infty} n^2\'

'Find the following limits.'

'(3) About real numbers'

"Let's summarize the key points of the methods we have learned so far for calculating the limits of sequences. Ways to find the limit of a sequence"

'Find the limit.'

'Find the following limit.'

"If f(x) is a differentiable function at x=a, then the following values can be expressed using a, f(a), f'(a), etc.:\n(1) lim₍ ₕ → 0₎ ( f(a + 3h) - f(a + h) ) / h \n(2) lim₍ ₓ → a₎ 1 / (x² - a²) { f(a) / x - f(x) / a }"

'Find the following limits. (1) \\( \\lim _{x \\rightarrow \\frac{\\pi}{2}} \\frac{1-\\sin x}{(2 x-\\pi)^{2}} \\) (2) \ \\lim _{x \\rightarrow 1} \\frac{\\sin \\pi x}{x-1} \ (3) \ \\lim _{x \\rightarrow \\infty} x \\sin \\frac{1}{x} \'

'Let f(x) be a function that is differentiable for all real numbers x and satisfies the following two conditions.'

'Find the following limit'

'Find the limit of the following sequences'

'(1) Find the following limits: (a) $\\lim _{x \\to-\\infty} \\frac{4^{x}}{3^{x}-2^{x}}$ (b) $\\lim _{x \\to \\infty}(3^{x}+2^{x})^{\\frac{1}{x}}$ (c) $\\lim _{x \\to \\infty}\\left\\{\\frac{1}{2} \\log _{3} x+\\log _{3}(\\sqrt{2 x+1}-\\sqrt{2 x-1})\\right\\}$ (2) For $x>1$, the inequality $0<\\log x<x$ holds. Using this inequality, find the limit $\\lim _{x \\to \\infty}\\frac{\\log x}{x}$. where $\\log x$ is the logarithm with base $e=2.71828 ....$.'

'Find the following limits.'

'For a series of positive terms $\\sum_{n=1}^{\\infty} a_{n}$, if $\\lim _{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_{n}}=k$, then if $k<1$ the series $\\sum_{n=1}^{\\infty} a_{n}$ converges, and if $k>1$ the series $\\sum_{n=1}^{\\infty} a_{n}$ diverges.'

'Find the following limits.'

'Solve problems related to the limit of irrational expressions.'

"Using L'Hopital's rule, find the following limits."

'Conditions for Existence of Limit'

'Find the following limits.'

'Find the following limits:'

'Find the limit of the sequence represented by the following expressions.'

'Find the limit of the following sequences'

'Find the following limits.'

"When f(x)=1/(1+x^2), the limit of (f(3x)-f(sin(x)))/x as x approaches 0 is greater than . Therefore, f'(0) = ."

'Find the following limits.'

'(2) Calculate \\( \\lim _{x \\rightarrow 0} \\frac{3 x^{2}}{\\sin ^{2} x}=\\lim _{x \\rightarrow 0} 3\\left(\\frac{x}{\\sin x}\\right)^{2} \n\\[=3 \\lim _{x \\rightarrow 0}\\left(\\frac{x}{\\sin x}\\right)^{2}=3 \\cdot 1^{2}=3\\]'

'Inside a circle C with radius 1 centered at O, there is a unique point A that is not at the center. Let the intersection point of the half-line OA and C be P0, and let P0 be the starting point to divide the circumference of C into n equal parts counterclockwise in order as P0, P1, P2, ..., Pn=P0. Define the distance between A and Pk as APk. Find the limit as n approaches infinity of 1/n * Σ(k=1 to n)(APk^2)^2. Given that OA=a. [Gunma University]'

'Find the limit of the sequence {r^n / n^k}.'

'Find the following limit.'

'Let {an} be an infinite sequence. A convergent value α converges (does not converge) to divergence infinity \ \\lim_{n \\rightarrow \\infty} \\alpha_{n}=\\infty \ or negative infinity \ \\lim_{n \\rightarrow \\infty} \\alpha_{n}=-\\infty \, it oscillates. When the limit of the sequence is \ \\infty \ or \ -\\infty \, it is not called a limit value.'

'(3) Find \\( \\lim_{x \\rightarrow 0} \\frac{\\sin \\left(x^{2}\\right)}{1-\\cos x} \\).'

'When r is a real number, find the limit lim(n→∞) (r^(2n+1))/(2+r^(2n)).'

'Find the limit of the sequence { r^n }.'

'Find the following limit. (Nihon Joshi Daigaku) \\n(4) \\( \\lim _{n \\rightarrow \\infty}\\left(\\frac{n+1}{n^{2}} \\log \\frac{n+1}{n}+\\frac{n+2}{n^{2}} \\log \\frac{n+2}{n}+\\cdots \\cdots+\\frac{n+n}{n^{2}} \\log \\frac{n+n}{n}\\right) \\)'

'(1) Prove $\\lim_{n \\to \\infty} \\frac{1}{n}\\left(\\sum_{k=n+1}^{2 n} \\log k-n \\log n\\right)=\\int_{1}^{2} \\log x dx$. (2) Find $\\lim_{n \\to \\infty}\\left\\{\\frac{(2 n)!}{n!n^{n}}\\right\\}^{\\frac{1}{n}}$.'

'Find the following limit.'

'Find the following limit. (3) $\\lim _{n \\rightarrow \\infty}\\left(\\frac{1}{n^{2}+1^{2}}+\\frac{2}{n^{2}+2^{2}}+\\frac{3}{n^{2}+3^{2}}+\\cdots \\cdots+\\frac{n}{n^{2}+n^{2}}\\right)$'

'Find the limit of the expression for the nth term.'

'Consider the sequences {an}, {bn}, is the following statement true? Prove it if true, or provide a counterexample if false. Where α, β are constants.'

'Let a_{n} = ∫_{n}^{n+1} 1/x dx, then lim_{n→∞} e^{n a_{n}} = .'

"Let a be a constant, and let the function f(x) be differentiable at x=a. Express the following limits using a and f'(a)."

'(3) As \ x \\longrightarrow \\infty \, \ \\frac{1}{x} \\longrightarrow 0 \, so \\\lim _{x \\rightarrow \\infty} \\tan \\frac{1}{x}=0\'

'Find the next limit of PR.'

'Find the following limits.\n1) \ \\lim _{n \\rightarrow \\infty} \\sum_{k=1}^{n} \\frac{n}{k^{2}+n^{2}} \\n2) \ \\lim _{n \\rightarrow \\infty} \\frac{\\pi}{n} \\sum_{k=1}^{n} \\cos ^{2} \\frac{k \\pi}{6 n} \\n3) \\( \\lim _{n \\rightarrow \\infty} \\sum_{k=1}^{n} \\frac{n^{2}}{(k+n)^{2}(k+2 n)} \\)\n4) \ \\lim _{n \\rightarrow \\infty} \\sum_{k=n+1}^{2 n} \\frac{n}{k^{2}+3 k n+2 n^{2}} \'

'Find the following limit: \\(\\lim _{n \\rightarrow \\infty}\\left(\\sqrt{n^{2}+2 n+2}-\\sqrt{n^{2}-n}\\right)\\)'

'Find the limit of the trigonometric function sin(x)/x as x approaches 0.'

'When r>-1, find the limit lim_{n \\rightarrow \\infty} \\frac{r^{n}}{2+r^{n+1}}. (2) When r is a real number, find the limit lim_{n \\rightarrow \\infty} \\frac{r^{2 n+1}}{2+r^{2 n}}. HINT (2) When r=-1, r^{2 n}=(-1)^{2 n}=\\left\\{(-1)^{2}\\right\\}^{n}=1^{n}=1. (1) When |r|<1, lim_{n \\rightarrow \\infty} r^{n}=0, lim_{n \\rightarrow \\infty} r^{n+1}=0, therefore lim_{n \\rightarrow \\infty} \\frac{r^{n}}{2+r^{n+1}}=\\frac{0}{2+0}=0. When r=1, r^{n}=r^{n+1}=1, therefore lim_{n \\rightarrow \\infty} \\frac{r^{n}}{2+r^{n+1}}=\\frac{1}{2+1}=\\frac{1}{3}. When r>1, \\left|\\frac{1}{r}\\right|<1, so lim_{n \\rightarrow \\infty}\\left(\\frac{1}{r}\\right)^{n+1}=0, therefore lim_{n \\rightarrow \\infty} \\frac{r^{n}}{2+r^{n+1}}=lim_{n \\rightarrow \\infty} \\frac{\\frac{1}{r}}{2\\left(\\frac{1}{r}\\right)^{n+1}+1}=\\frac{\\frac{1}{r}}{2 \\cdot 0+1}=\\frac{1}{r}'

'Find the limit of the sequence represented by the following expressions:'

'Limit of the sequence {r ^ n}'

'Find the limit of the given basic example 35 (3).'

'Chapter 2\nLimit\nEX Sequence \ \\{a_{n}\\} \ satisfies \\( a_{n}>0(n=1,2, \\cdots) \\), \ \\lim_{n \\rightarrow \\infty} \\frac{-5a_{n}+3}{2a_{n}+1}=-1 \, find \ \\lim_{n \\rightarrow \\infty} a_{n} \'

'Find the following limit. \ \\lim _{x \\rightarrow -0} \\frac{\\sqrt{1-\\cos x}}{x} \'

'Please provide a proof using the method of infinite descent.'