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Find the parameter making left limit, right limit and value agree at one joining point

This type of question, asked 9 times: 11 May 2023 S2, 9 May 2024 S2, 11 May 2024 S2, 12 May 2024 S2, 19 Apr 2025 S1, 20 Apr 2025 S1, 22 Apr 2025 S2, 20 Apr 2026 S2, 19 May 2026 S1. All 9 questions below, oldest first.

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  1. Question 1

    11 May 2023 S2

    131
    Let f(x)={1−cos⁡4xx2,x<0a,x=0x16+x−4,x>0f(x) = \begin{cases}\frac{1-\cos 4x}{x^2}, & x<0\\ a, & x=0\\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x>0\end{cases}. If f(x)f(x) is continuous at x=0x=0, then the value of a is

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  2. Question 2

    9 May 2024 S2

    144
    f(x)={1−cos⁡kxx2,x≤0x16+x−4,x>0f(x) = \begin{cases}\dfrac{1-\cos kx}{x^2}, & x \leq 0 \\ \dfrac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x > 0\end{cases} is continuous at x=0x=0, then the value of kk is

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  3. Question 3

    11 May 2024 S2

    102
    If f(x)={x−3∣x−3∣+a,x<3a+b,x=3∣x−3∣x−3+b,x>3f(x) = \begin{cases} \frac{x-3}{|x-3|}+a, & x<3 \\ a+b, & x=3 \\ \frac{|x-3|}{x-3}+b, & x>3 \end{cases} is continuous at x=3x=3, then the value of a−ba-b is:

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  4. Question 4

    12 May 2024 S2

    120
    Given f(x)={1−cos⁡4xx2,x<0α,x=0x16+x−4,x>0f(x)=\begin{cases}\dfrac{1-\cos4x}{x^2}, & x<0\\ \alpha, & x=0\\ \dfrac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x>0\end{cases}. If f(x)f(x) is continuous at x=0x=0, then value of α\alpha is

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  5. Question 5

    19 Apr 2025 S1

    146
    If f(x)f(x) is continuous at point x=0x= 0 where f(x)={3sin⁡x+5tan⁡xax−1,x<02log⁡2,x=08x+2xcosx bx−1,x>0f(x) =\left\{ \begin{matrix} \frac{3\sin x+ 5\tan x}{a^{x}- 1} & ,x< 0 \\ \frac{2}{\log2} & ,x= 0 \\ \frac{8x+ 2xcosx}{{\text{ }b}^{x}- 1} & ,x> 0 \end{matrix} \right. then the values of aa and bb, respectively, are .

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  6. Question 6

    20 Apr 2025 S1

    122
    If f(x)={mx+1,x⩽π2sin⁡x+n,x>π2f(x) =\left\{ \begin{matrix} mx+ 1, & x\leqslant \frac{\pi}{2} \\ \sin x+ n, & x>\frac{\pi}{2} \end{matrix} \right., is continuous at x=π2,( m,n∈Z)x=\frac{\pi}{2},(\text{ }m,n \in\mathbb{Z}) then

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  7. Question 7

    22 Apr 2025 S2

    137
    If f(x)={1−cos⁡4xx2,x<0a,x=0x(16+x)1/2−4,x>0f(x)=\begin{cases}\dfrac{1-\cos4x}{x^2}, & x<0\\ a, & x=0\\ \dfrac{\sqrt{x}}{(16+\sqrt{x})^{1/2}-4}, & x>0\end{cases} is continuous at x=0x=0, then a=a=

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  8. Question 8

    20 Apr 2026 S2

    69
    If the function f(x)f(x) defined by f(x)={ax+1if x≤3bx+3if x>3f(x) = \begin{cases} ax + 1 & \text{if } x \le 3 \\ bx + 3 & \text{if } x > 3 \end{cases} is continuous at x=3x = 3, then (a−b)=(a - b) = ..........

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  9. Question 9

    19 May 2026 S1

    71
    If f(x)={1−sin⁡5x5cos⁡2x,x<π2a,at x=π2b(1−sin⁡x)8(π−2x)2,x>π2f(x) = \begin{cases} \dfrac{1 - \sin^5 x}{5\cos^2 x} & , x < \dfrac{\pi}{2} \\ a & , \text{at } x = \dfrac{\pi}{2} \\ \dfrac{b(1 - \sin x)}{8(\pi - 2x)^2} & , x > \dfrac{\pi}{2} \end{cases} is continuous at x=π2x = \dfrac{\pi}{2}, then (ba)53=…\left(\dfrac{b}{a}\right)^{\frac{5}{3}} = \ldots

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