Answer to Question #177817 in Real Analysis for Nikhil

Question

The limit: limit x→0^+ (xcosecx)^x does not exist

True or false with full explanation

Answer 1

Let us show that the limit $\lim\limits_{ x\to 0^+} (x\cosec x)^x$ exists.

Since $\lim\limits_{ x\to 0} \frac{\sin x}{x}=1,$ we conclude that

$\lim\limits_{ x\to 0^+} (x\cosec x)^x=\lim\limits_{ x\to 0^+} (\frac{x}{\sin x})^x= \lim\limits_{ x\to 0^+} (\frac{\sin x}{ x})^{-x}=1^0=1$

Answer: false