Answer to Question #177817 in Real Analysis for Nikhil

Question #177817

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

True or false with full explanation



1
Expert's answer
2021-04-13T13:37:09-0400

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


Answer: false


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