Question

Evaluate the following limits

$lim \: x \to \: 0 \: \: \: {(sinx)}^{x}$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm :\longmapsto\:\displaystyle\lim_{x \to 0} \: {(sinx)}^{x}$

Let assume that,

$\rm :\longmapsto\:y = \displaystyle\lim_{x \to 0} \: {(sinx)}^{x}$

Taking log on both sides, we get

$\rm :\longmapsto\:logy = \displaystyle\lim_{x \to 0} \: log {(sinx)}^{x}$

We know,

$\boxed{ \tt{ \: log {x}^{y} = y \: logx \: }}$

So, using this identity,

$\rm :\longmapsto\:logy = \displaystyle\lim_{x \to 0} \: x \: log(sinx)$

can be rewritten as

$\rm :\longmapsto\:logy = \displaystyle\lim_{x \to 0} \: \frac{logsinx}{ \dfrac{1}{x} }$

We know,

$\boxed{ \tt{ \: \dfrac{d}{dx}logx = \frac{1}{x}}}$

and

$\boxed{ \tt{ \: \dfrac{d}{dx} \frac{1}{ {x}^{n} } = \frac{ - n}{ {x}^{n + 1} }}}$

So, using these, we get

$\rm :\longmapsto\:logy = \displaystyle\lim_{x \to 0} \frac{ \dfrac{1}{sinx} \dfrac{d}{dx}sinx}{ \dfrac{ - 1}{ {x}^{2} } }$

$\rm :\longmapsto\:logy = \displaystyle\lim_{x \to 0} \frac{ \dfrac{1}{sinx} \times cosx}{ \dfrac{ - 1}{ {x}^{2} } }$

$\rm :\longmapsto\:logy = \displaystyle\lim_{x \to 0} \frac{cotx}{ \dfrac{ - 1}{ {x}^{2} } }$

$\rm :\longmapsto\:logy = \displaystyle\lim_{x \to 0} \: {x}^{2} \: cotx$

$\rm :\longmapsto\:logy = \displaystyle\lim_{x \to 0} \: \frac{ {x}^{2} }{tanx}$

$\rm :\longmapsto\:logy = \displaystyle\lim_{x \to 0} \: \frac{x }{tanx} \times x$

$\rm :\longmapsto\:logy = 1 \times 0$

$\rm :\longmapsto\:logy = 0$

$\bf\implies \:y = {e}^{0} = 1$

Hence,

$\bf\implies \:\:\boxed{ \tt{ \: \displaystyle\lim_{x \to 0} \: {(sinx)}^{x} = e \: \: }}$

More to know :-

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}$