Let f : [-π/2, π/2 ] rightarrow R be a differentiable function such that f(0)=1/2 , If the lim x rightarrow 0 frac x int 0 x f(t) d t e x 2 -1 =α , then 8 α 2 is equal to :

Mathematics · Definite Integration

JEE Main 2024 — 30 January, Shift 1 — Question 20

Let $\mathrm{f}:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$ , If the $\lim _{x \rightarrow 0} \frac{x \int_{0}^{x} f(t) d t}{e^{x^{2}}-1}=\alpha$ , then $8 \alpha^{2}$ is equal to :

Option A:
16
Option B:
2
Correct
Option C:
1
Option D:
4

Answer: B

Step-by-step solution

$\lim _{x \rightarrow 0} \frac{x \int_{0}^{x} f(t) d t}{\left(\frac{e^{x^{2}}-1}{x^{2}}\right) \times x^{2}}$

$\lim _{x \rightarrow 0} \frac{\int_{0}^{x} f(t) d t}{x}$

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

$=\lim _{x \rightarrow 0} \frac{f(x)}{1}$ (using L Hospital)

$\mathrm{f}(0)=\frac{1}{2}$

$\alpha=\frac{1}{2}$

$8 \alpha^{2}=2$

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Exam: JEE Main 2024
Paper: 30 January, Shift 1
Subject: Mathematics
Chapter: Definite Integration
Topic: Leibnitz rule & its application in limits

Answer: B

Step-by-step solution

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