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1 to the power infinity form: limit, or value making the function continuous

This type of question, asked 7 times: 3 May 2023 S2, 10 May 2023 S2, 22 Apr 2025 S1, 22 Apr 2025 S2, 11 Apr 2026 S1, 19 Apr 2026 S2, 15 May 2026 S2. All 7 questions below, oldest first.

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  1. Question 1

    3 May 2023 S2

    104
    If the function f(x)=(5x−88−3x)32x−4f(x)=\left(\dfrac{5x-8}{8-3x}\right)^{\frac{3}{2x-4}} if x≠2x\neq2, =k=k if x=2x=2, is continuous at x=2x=2, then k=k=

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  2. Question 2

    10 May 2023 S2

    129
    If f(x)=(1+tan⁡x1+sin⁡x)csc⁡xf(x) = \left(\frac{1+\tan x}{1+\sin x}\right)^{\csc x} is continuous at x=0x=0, then f(0)f(0) is equal to

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  3. Question 3

    22 Apr 2025 S1

    134
    lim⁡x→∞(x+8x+1)x+5=…\lim_{x\rightarrow \infty} \left( \frac{x+ 8}{x+ 1} \right)^{x+ 5}= \ldots

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  4. Question 4

    22 Apr 2025 S2

    111
    lim⁡x→1(log⁡33x)log⁡x8=\lim_{x \rightarrow 1} \left( \log_{3}3x \right)^{\log_{x}8}=... .

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  5. Question 5

    11 Apr 2026 S1

    68
    Let the function f(x)f(x) be defined as: f(x)={[tan⁡(π4+x)]1x,x≠0k,x=0f(x) = \begin{cases} \left[\tan\left(\dfrac{\pi}{4} + x\right)\right]^{\frac{1}{x}}, & x \neq 0 \\ k, & x = 0 \end{cases} If f(x)f(x) is continuous at x=0x = 0, then the value of kk is...

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  6. Question 6

    19 Apr 2026 S2

    70
    If the function f(x)=(5x−88−3x)32x−4f(x) = \left(\dfrac{5x - 8}{8 - 3x}\right)^{\frac{3}{2x - 4}}, for x≠2x \neq 2 is continuous at x=2x = 2, then the value of f(2)f(2) is...

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  7. Question 7

    15 May 2026 S2

    63
    If lim⁡x→0(f(x)x2)\lim_{x \to 0}\left(\dfrac{f(x)}{x^2}\right) exists finitely and lim⁡x→0(1+x+f(x)x)1/x=e3\lim_{x \to 0}\left(1 + x + \dfrac{f(x)}{x}\right)^{1/x} = e^3, then ∫f(x)⋅log⁡x dx\int f(x) \cdot \log x\, dx is equal to:

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