If the function f(x) = ((4^x-1)^3)/(/4*(1+x^2/3)) is continuous at x = 0, then the value of f(0) is: (Take _e x = x) Option 1: 1/12( 4)^3 Option 2: 12( 4)^3 Option 3: 4 Option 4: 12

Answer: 12( 4)^3. Solution: For continuity f(0) equals the limit. Using 4^x-1 x 4, /4 x/4 and (1+x^2/3) x^2/3, the expression tends to (x^3( 4)^3)/(x^3/12) = 12( 4)^3.

Calculus

Continuity & Differentiability

Hard

core

If the function $f(x) = \dfrac{(4^x-1)^3}{\sin\dfrac{x}{4}\cdot\log\left(1+\dfrac{x^2}{3}\right)}$ is continuous at $x = 0$ , then the value of $f(0)$ is: (Take $\log_e x = \log x$ )

$\dfrac{1}{12}(\log 4)^3$

$12(\log 4)^3$

$\log 4$

✅ Correct Option: 2

For continuity $f(0)$ equals the limit. Using $4^x-1 \approx x\log 4$ , $\sin\frac{x}{4} \approx \frac{x}{4}$ and $\log\left(1+\frac{x^2}{3}\right) \approx \frac{x^2}{3}$ , the expression tends to $\frac{x^3(\log 4)^3}{x^3/12} = 12(\log 4)^3$ .

2025: 13 May Shift 1

Match List-I with List-II

List-I	List-II
(Function)	(Derivative with respect to 'x')
(A) $f(x) = x^x$	(I) $ax^{a-1}$
(B) $f(x) = a^x$	(II) 0
(C) $f(x) = a^a$	(III) $a^x log_e a$
(D) $f(x) = x^a$	(IV) $x^x(1 + log_e x)$

Choose the correct answer from the options given below:

2026: 15 May Shift 1

If $f(x) = \begin{cases} \dfrac{1}{|x|}, & x \geq 1 \\ ax^2+b, & x < 1 \end{cases}$ is differentiable at $x=1$ then

2026: 19 May Shift 1

At $x = 2$ , $f(x) = [x]$ is

[where [.] is greatest integer function]

2023: 15 June Shift 2

If $y = \log\left[\frac{x^2}{e^2}\right]$ then value of $\frac{d^2y}{dx^2}$ is :

2025: 16 May Shift 1

Match List-I with List-II

List-I	List-II
(A) $f(x) = x \sin x$	(I) is not continuous at $x = -3$
(B) $f(x) = \frac{\vert x\vert }{x}, x \neq 0$ and $f(x) = 1 \text{ at } x = 0$	(II) is continuous everywhere
(C) $f(x) = x - [x]$ , $[x]$ denotes greatest integer function	(III) is not differentiable at $x = 1$
(D) $f(x) = e^{\vert x - 1\vert }$	(IV) is not continuous at $x = 0$

Choose the correct answer from the options given below:

2026: 21 May Shift 2

If $y = x\log_e\left(\dfrac{x}{2+x}\right)$ , then $x^3\dfrac{d^2y}{dx^2}$ is equal to:

If the function f(x)=(4x−1)3sin⁡x4⋅log⁡(1+x23)f(x) = \dfrac{(4^x-1)^3}{\sin\dfrac{x}{4}\cdot\log\left(1+\dfrac{x^2}{3}\right)}f(x)=sin4x​⋅log(1+3x2​)(4x−1)3​ is continuous at x=0x = 0x=0, then the value of f(0)f(0)f(0) is: (Take log⁡ex=log⁡x\log_e x = \log xloge​x=logx)

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If the function $f(x) = \dfrac{(4^x-1)^3}{\sin\dfrac{x}{4}\cdot\log\left(1+\dfrac{x^2}{3}\right)}$ is continuous at $x = 0$ , then the value of $f(0)$ is: (Take $\log_e x = \log x$ )