Question

$\underset{x\to \pi }{\lim }\frac{|x+\pi |}{\sin x}$

• A. exist and is equal to 1
• B. does not exist and LHL = 1
• C. does not exist and RHL = 1
• D. does not exist and LHL is not finite

Answer 1

Answer: (C)

$\underset{x\to -\pi }{\lim }\frac{|x+\pi |}{\sin x}=\underset{t\to 0}{\lim }\frac{|t|}{\sin (t-\pi )}$
Put $x+\pi =t=\underset{t\to 0}{\lim }\frac{|t|}{-\sin t}$
LHL imit = $\underset{t\to 0^{-}}{\lim }\frac{-t}{-\sin t}=1$ ,
RHL imit = $\underset{t\to 0^{+}}{\lim }\frac{t}{-\sin t}=-1$.
$\therefore$ Limit does not exist.