First slide

Introduction to limits

Question

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

Moderate

A

$\frac{8}{π} f (2)$
B

$\frac{2}{π} f (2)$
C

$\frac{2}{π} f (\frac{1}{2})$
D

$4 f (2)$

Solution

$lim_{x \to π / 4} \frac{\int_{x^{2}}^{\sec^{2} x} f (t) d t}{x^{2} - π^{2} / 16}$
$\begin{array}{l} = lim_{x \to π / 4} \frac{2 \sec^{2} x \tan x f (\sec^{2} x)}{2 x} [\begin{array}{c} Using Leibniz and \\ L' Hospital's rules \end{array}] \\ = \frac{\sec^{2} \frac{π}{4} \tan \frac{π}{4} f (\sec^{2} \frac{π}{4})}{(π / 4)} = \frac{8}{π} f (2) \end{array}$

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