lim x 4 2 sec 2 x f ( t ) d t x 2 - 2 16 equals [2007]

Question

$\lim_{x \to \frac{π}{4}} \frac{\int_{2}^{\sec^{2} x} f (t) d t}{x^{2} - \frac{π^{2}}{16}}$ equals [2007]

Smart Achievers · Accepted Answer

(1)
$\lim_{x \to \frac{π}{4}} \frac{\int_{2}^{\sec^{2} x} f (t) d t}{x^{2} - \frac{π^{2}}{16}}$ $[\frac{0}{0} -form]$

$= \lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [\int_{2}^{\sec^{2} x} f (t) d t]}{\frac{d}{d x} (x^{2} - \frac{π^{2}}{16})}$ (Using L'Hospital rule)

$= \lim_{x \to \frac{π}{4}} \frac{f (\sec^{2} x) \cdot 2 \sec^{2} x \tan x}{2 x}$

$[∵ \frac{d}{d x} [\int_{g (x)}^{h (x)} f (t) d t] = f (h (x)) h' (x) - f (g (x)) g' (x)]$

$= \frac{f (2) \times 2 \times 2 \times 1}{2 \times \frac{π}{4}} = \frac{8}{π} f (2)$

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