lim _x arrow 0 1/x ∫_0^x(1+sin 3 t)^1/tdt is equal to

Correct answer: (D) e^3. lim _x arrow 0 frac∫_0^x(1+sin 3 t)^1/t d tx Differentiating numerator and denominator L=lim _x arrow 0 frac(1+sin 3 x)^1/x1 ⇒ L =e^lim _x arrow 0 1/x sin 3 x ⇒ L=e^lim _x arrow 0 (3 sin 3 x)/(3 x) ⇒ L =e^3 × 1=e^3

Q68 / 110 · VITEEE 2024 · Mathematics

$\lim _{x \rightarrow 0} \frac{1}{x} \int_0^x(1+\sin 3 t)^{\frac{1}{t}}dt$ is equal to

A
2
B
1
C
$e^2$
Correct answer
D
$e^3$

Solution

$\lim _{x \rightarrow 0} \frac{\int_0^x(1+\sin 3 t)^{\frac{1}{t}} d t}{x}$ Differentiating numerator and denominator $\begin{aligned} & & L=\lim _{x \rightarrow 0} \frac{(1+\sin 3 x)^{\frac{1}{x}}}{1} \\ \Rightarrow & L & =e^{\lim _{x \rightarrow 0} \frac{1}{x} \sin 3 x} \Rightarrow L=e^{\lim _{x \rightarrow 0} \frac{3 \sin 3 x}{3 x}} \\ \Rightarrow & L & =e^{3 \times 1}=e^3 \end{aligned}$

Hence, the correct answer is option: 'D'.

$\lim _{x \rightarrow 0} \frac{1}{x} \int_0^x(1+\sin 3 t)^{\frac{1}{t}}dt$ is equal to

Solution

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