The value of lim x - 01x sin x - where x - 0is

Lim _x → 0[ (1x )^sin x ] For nbsp; lim_x → 0 (1x )^sin x=L , take log both sides . log L= lim_x → 0 sin x log( 1x nbsp; nbsp;) ⇒ L= e^lim_x → 0 log x ...

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Standard XII

Mathematics

Single Point Continuity

The value of ...

Question

The value of $lim x \to 0 ({(\frac{1}{x})}^{sin x})$ , where $x > 0$ is

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Solution

$lim x \to 0 [{(\frac{1}{x})}^{sin x}]$

For
$lim x \to 0 {(\frac{1}{x})}^{sin x} = L$ , take log both sides .
$log L = lim x \to 0 sin x log (\frac{1}{x})$
$\Rightarrow$ $L = e^{lim x \to 0 \frac{- log x}{cosec x}}$

$= e^{lim x \to 0 \frac{\frac{- 1}{x}}{- cosec x cot x}} [Using L 'Hospita's rule]$

$= e^{lim x \to 0 \frac{sin x}{x} tan x} = e^{0} = 1$

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Single Point Continuity

Standard XII Mathematics

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The value of limx→0((1x)sin x), where x>0 is

limx→0[(1x)sin x] For limx→0(1x)sin x=L , take log both sides . logL=limx→0sinxlog(1x) ⇒ L=elimx→0−logxcosecx =elimx→0−1x−cosec x cot x [Using L 'Hospita's rule] =elimx→0sin xxtan x=e0=1

The value of $lim x \to 0 ({(\frac{1}{x})}^{sin x})$ , where $x > 0$ is