Explanation

(c)
$\lim_{x \rightarrow 0}\left( \frac{\int_{0}^{x^{2}}{\sec^{2}{t\ dt}}}{x\sin x} \right) = \lim_{x \rightarrow 0}{\ \ \frac{\sec^{2}\ x^{2}.2x}{\sin x + x\cos x}}$
$= \lim_{x \rightarrow 0}{\ \ \frac{{2x.\sec}^{2}\ x^{2}}{x\left( \frac{\sin x}{x} + \cos x \right)}}$
$= \frac{2 \times 1}{1 + 1} = 1$ $\left\lbrack \because\ \lim_{x \rightarrow 0}{\ \frac{\sin x}{x} = 1} \right\rbrack$