Question

Solve the equation:-
$\stackrel{lim}{x\to \pi }\frac{1+{\sec }^{3}x}{{\tan }^{2}x}$

Answer 1

${lim}_{x\to \pi }\frac{1+{\sec }^{3}x}{{\tan }^{2}x}$

$={lim}_{x\to \pi }\frac{(1+\sec x)(1+{\sec }^{2}x-\sec x)}{{\sec }^{2}x-1}$

$={lim}_{x\to \pi }\frac{(1+\sec x)(1+{\sec }^{2}x-\sec x)}{(\sec x+1)(\sec x-1)}$

$={lim}_{x\to \pi }\frac{(1+{\sec }^{2}x-\sec x)}{(\sec x-1)}$

$=\frac{(1+{\sec }^2\pi -\sec \pi )}{(\sec \pi -1)}$

$=\frac{1+1+1}{-1-1}=-\frac{3}{2}$