Question

Solve: ${lim}_{x\to 0+}(\sin x{)}^{\tan x}$

• A. $e^3$
• B. $e^2$
• C. $-1$
• D. $1$

Answer 1

The correct option is D $1$

$\underset{x\to 0^{+}}{lim}(\sin x{)}^{\tan x}=0^o$ form

$\log L=\underset{x\to 0^{+}}{lim}\tan \log \sin x=\underset{x\to 0}{lim}\frac{\log \sin x}{\cot x}=\frac{\infty }{\infty }$

$\log L=\underset{x\to 0}{lim}\frac{\frac{\cos x}{\sin x}}{cose{c}^{2}x}=\underset{x\to 0}{lim}\sin x\cos x$

$\log L=0$

$\Rightarrow L=e^o=1$