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Limit of the 1 to the power infinity form

This type of question, asked 8 times: 2021 Paper 3, 28 Jul 2022, 22 Jan 2025, 23 Jan 2025, 29 Jan 2025, 3 Apr 2025, 4 Apr 2025, 21 Jan 2026 S1. All 8 questions below, oldest first.

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  1. Question 1

    2021 Paper 3

    76
    lim⁡n→∞(1+1+12+……+1nn2)n\lim_{n \rightarrow \infty} \left( 1 +\frac{1 +\frac{1}{2}+ \ldots\ldots +\frac{1}{n}}{n^{2}} \right)^{n} is equal to:

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  2. Question 2

    28 Jul 2022

    89
    lim⁡x→0((x+2cos⁡x)3+2(x+2cos⁡x)2+3sin⁡(x+2cos⁡x)(x+2)3+2(x+2)2+3sin⁡(x+2))100x\underset{x\rightarrow 0}{\lim} \left( \frac{(x+ 2\cos x)^{3}+ 2(x+ 2\cos x)^{2}+ 3\sin(x+ 2\cos x)}{(x+ 2)^{3}+ 2(x+ 2)^{2}+ 3\sin(x+ 2)} \right)^{\frac{100}{x}}is equal to

  3. Question 3

    22 Jan 2025

    132
    If lim⁡x→∞((e1−e)(1e−x1+x))x=α\lim_{x \rightarrow \infty} \left( \left( \frac{e}{1 - e} \right)\left( \frac{1}{e}-\frac{x}{1 + x} \right) \right)^{x}= \alpha, then the value of log⁡eα1+log⁡eα\frac{\log_{e}\alpha}{1 +\log_{e}\alpha} equals :

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  4. Question 4

    23 Jan 2025

    145
    lim⁡x→∞(2x2−3x+5)(3x−1)x2(3x2+5x+4)(3x+2)x\lim_{x \rightarrow \infty} \frac{\left( 2x^{2}- 3x + 5 \right)(3x - 1)^{\frac{x}{2}}}{\left( 3x^{2}+ 5x + 4 \right)\sqrt{(3x + 2)^{x}}} is equal to:

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  5. Question 5

    29 Jan 2025

    147
    If lim⁡t→0(∫01(3x+5)tdx)1t=α5e(85)23\lim_{t \rightarrow 0} \left( \int_{0}^{1} (3x + 5)^{t}dx \right)^{\frac{1}{t}}=\frac{\alpha}{5e}\left( \frac{8}{5} \right)^{\frac{2}{3}}, then α\alpha is equal to ____\_\_\_\_ .

  6. Question 6

    3 Apr 2025

    148
    If Limx→0(tan⁡xx)1x2=p{Lim}_{x \rightarrow 0}\left( \frac{\tan x}{x} \right)^{\frac{1}{x^{2}}} = p, then 96log⁡ep96\log_{e}p is equal to.

  7. Question 7

    4 Apr 2025

    127
    Let ff be a differentiable function on RR such that f(2)=1f\left( 2 \right)= 1, f′(2)=4f^{'}\left( 2 \right)= 4. Let lim⁡x→0(f(2+x))3x=eα\lim_{x\rightarrow 0} (f(2 + x))^{\frac{3}{x}}=e^{\alpha}. Then the number of times the curve y=4x3−4x2−4(α−7)x−αy= 4x^{3}- 4x^{2}- 4(\alpha- 7)x-\alpha meets xx-axis is.

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  8. Question 8

    21 Jan 2026 S1

    60
    Let f:R→(0,∞)f:R \rightarrow (0,\infty) be a twice differentiable function such that f(3)=18,f′(3)=0f(3) = 18,f^{'}(3) = 0 and f′′(3)=4f^{''}(3) = 4. Then lim⁡x→1(log⁡e(f(2+x)f(3))18(x−1)2)\lim_{x \rightarrow 1} \left( \log_{e}\left( \frac{f(2 + x)}{f(3)} \right)^{\frac{18}{(x - 1)^{2}}} \right) is equal to :

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