Determine-lim-x-0-sin-x-1-x-

Question Number 4441 by Rasheed Soomro last updated on 27/Jan/16

Determine

\lim_{x \to 0} {(\sin x)}^{1 / x} .

Commented by Yozzii last updated on 29/Jan/16

T h e l i m i t d o e s n o t e x i s t .

l = \lim_{x \to 0^{-}} {(s i n x)}^{1 / x}

l n l = \lim_{x \to 0^{-}} \frac{1}{x} l n s i n x .

F o r x \to 0^{-}, l n s i n x \in C s i n c e s i n x < 0 a n d \frac{1}{x} \to - \infty .

∴ l n l \in C \Rightarrow l \in C . B u t, \lim_{x \to 0^{+}} {(s i n x)}^{1 / x} = 0 .

Commented by Jens last updated on 29/Jan/16

P u t y = 1 / x . T h e n s i n x = s i n (1 / y)

\Rightarrow 1 / y a s y \Rightarrow \infty s o {(s i n x)}^{1 / x} \Rightarrow

{(1 / y)}^{y} = 1 / y^{y} \Rightarrow 0 f o r y \Rightarrow \infty s o t h e

l i m i t d o e s e x i s t!

Commented by Yozzii last updated on 30/Jan/16

F o r a l i m i t o f f (x) t o e x i s t n e a r x = a w e n e e d t h a t

\lim_{x \to a^{+}} f (x) = \lim_{x \to a^{-}} f (x) .

B y l e t t i n g x = 1 / y \Rightarrow y = 1 / x

∴ a s x \to 0^{-} \Rightarrow y \to - \infty

a s x \to 0^{+} \Rightarrow y \to \infty

I f t h e l i m i t e x i s t s t h e n w e r e q u i r e

\lim_{y \to - \infty} {(s i n (\frac{1}{y}))}^{y} = \lim_{y \to + \infty} {(s i n (\frac{1}{y}))}^{y}

B u t, w h i l e \lim_{y \to \infty} {(s i n (1 / y))}^{y} = {(s i n (+ 0))}^{\infty} = + 0

\lim_{y \to - \infty} {(s i n (1 / y))}^{y} = {(s i n (- 0))}^{- \infty} = \frac{1}{{(s i n (- 0))}^{\infty}} = \frac{1}{- 0}

w h i c h i s u n d e f i n e d .

Commented by Yozzii last updated on 30/Jan/16

Commented by Yozzii last updated on 30/Jan/16

F o r n e g a t i v e x n e a r z e r o,

n o r e a l v a l u e o f {(s i n x)}^{1 / x} e x i s t s .

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