_ x 4 _2^ ^2 x f(t) d t x^2- ^2 16 equals - Vidyadip

MCQ

$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}$ equals

✓
$\frac{8}{\pi} f(2)$
B
$\frac{2}{\pi} f(2)$
C
$\frac{2}{\pi} f\left(\frac{1}{2}\right)$
D
$4 \mathrm{f}(2)$

✓

Answer

Correct option: A.

$\frac{8}{\pi} f(2)$

a
$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}} \quad\left(\frac{0}{0} \text { form }\right)$

Let $L=\lim _{x \rightarrow \frac{\pi}{4}} \frac{f\left(\sec ^2 x\right) 2 \sec x \sec x \tan x}{2 x}$

$\therefore \mathrm{L}=\frac{2 \mathrm{f}(2)}{\pi / 4}=\frac{8 \mathrm{f}(2)}{\pi} \text {. }$

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