Let f(x)=int 0 x (t+sin (1-e t ) ) d t, x in mathbb R . Then lim x rightarrow 0 frac f(x) x 3 is equal to

Mathematics · Definite Integration

JEE Main 2024 — 4 April, Shift 2 — Question 8

Let $f(x)=\int_{0}^{x}\left(t+\sin \left(1-e^{t}\right)\right) d t, x \in \mathbb{R}$ . Then $\lim _{x \rightarrow 0} \frac{f(x)}{x^{3}}$ is equal to

$\lim _{x \rightarrow 0} \frac{f(x)}{x^{3}}$

{Using L Hopital Rule.}

$\lim _{x \rightarrow 0} \frac{f^{\prime}(x)}{3 x^{2}}=\lim _{x \rightarrow 0} \frac{x+\sin \left(1-e^{x}\right)}{3 x^{2}}$

Using L.H. Rule

$=\lim _{x \rightarrow 0} \frac{-\left[\sin \left(1-e^{x}\right)\left(-e^{x}\right) \cdot e^{x}+\cos \left(1-e^{x}\right) \cdot e^{x}\right]}{6}$

$=-\frac{1}{6}$

Answer key and solution verified before publishing.

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