Question

$\underset{x\to 0}{lim}\mid x{\mid }^{\sin x}=$

• A. $0$
• B. $-1$
• C. $1$
• D. none

Answer 1

The correct option is C $1$

$\underset{x\to 0}{lim}{\left|x\right|}^{\sin x}=l$ (say) [It is of the form $\left(0^0\right)$]

$l=\underset{x\to 0}{lim}{\left|x\right|}^{\sin x}$

Taking ${\log }_e$ on both sides,

$\ln \left(l\right)=\underset{x\to 0}{lim}\left(\sin x\ln \left|x\right|\right)$

$\ln \left(l\right)=\underset{x\to 0}{lim}\sin x\ln \left|x\right|$

$\ln \left(l\right)=0$

$l=e^0$

$l=1$