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Trigonometric limit as x tends to pi/4 (value of f(pi/4))

This type of question, asked 6 times: 3 May 2023 S1, 16 May 2023 S1, 16 May 2023 S2, 19 Apr 2026 S1, 12 May 2026 S2, 19 May 2026 S2. All 6 questions below, oldest first.

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  1. Question 1

    3 May 2023 S1

    103
    Let f(x)=1−tan⁡x4x−π, x≠π4, x∈ ⁣[0,π2]f(x)=\frac{1-\tan x}{4x-\pi},\ x\neq\frac{\pi}{4},\ x\in\!\left[0,\frac{\pi}{2}\right]. If f(x)f(x) is continuous in [0,π2]\left[0,\frac{\pi}{2}\right], then f ⁣(π4)f\!\left(\frac{\pi}{4}\right) is

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  2. Question 2

    16 May 2023 S1

    121
    Let f(x)=1−tan⁡x4x−π,  x≠π4,  x∈[0,π2)f(x) = \frac{1-\tan x}{4x-\pi},\; x \neq \frac{\pi}{4},\; x \in \left[0, \frac{\pi}{2}\right). f(x)f(x) is continuous in [0,π2)\left[0, \frac{\pi}{2}\right), then f ⁣(π4)f\!\left(\frac{\pi}{4}\right) is

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  3. Question 3

    16 May 2023 S2

    127
    If the function ff defined on (π6,π3)\left(\frac{\pi}{6}, \frac{\pi}{3}\right) by f(x)=2cos⁡x−1cot⁡x−1,x≠π4f(x) = \frac{\sqrt{2}\cos x - 1}{\cot x - 1}, x \neq \frac{\pi}{4}; f(x)=k,x=π4f(x) = k, x = \frac{\pi}{4} is continuous, then kk is equal to

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  4. Question 4

    19 Apr 2026 S1

    60
    If the function f(x)=22−(cos⁡x+sin⁡x)31−sin⁡2xf(x) = \dfrac{2\sqrt{2} - (\cos x + \sin x)^3}{1 - \sin 2x} is continuous at x=π4x = \dfrac{\pi}{4}, then the value of f(π4)f\left(\dfrac{\pi}{4}\right) is ...

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  5. Question 5

    12 May 2026 S2

    70
    If f(x)={tan⁡x−cot⁡xx−π4,x≠π4a,x=π4f(x) = \begin{cases} \dfrac{\tan x - \cot x}{x - \frac{\pi}{4}}, & x \neq \frac{\pi}{4} \\ a, & x = \frac{\pi}{4} \end{cases}, then the value of ′a′'a' so that f(x)f(x) is continuous at x=π/4x = \pi/4 is...

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  6. Question 6

    19 May 2026 S2

    58
    The value of lim⁡x→π4[4−(cos⁡x+sin⁡x)41−sin⁡2x]\lim_{x \to \frac{\pi}{4}}\left[\dfrac{4 - (\cos x + \sin x)^4}{1 - \sin 2x}\right] is equal to

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