Limit with L'Hospital with infinite indeterminate formats I'm trying to find the limit:
$$\large \lim_{x\to0}(\sin x)^x$$
Whst I did was apply L'Hospital Rule:
$$\large \text{let }y =(\sin x)^x\implies
\ln y=x\ln\sin x$$
$$\large 
\lim_{x\to0}\ln y
=
\lim_{x\to0} x\ln\sin x
=
\lim_{x\to0}\frac x{\frac1{\ln\sin x}}
=
\lim_{x\to0} \frac1{\frac {-\cos x}{(\ln\sin x)^2\sin x}}
=
\lim_{x\to0} (-\tan x )(\ln\sin x)^2
=
\lim_{x\to0} \frac{(\ln\sin x)^2}{\frac1{(-\tan x )}}
=
...
$$
Ultimately it keeps on going, please help me.

Hint: $$\sin x\sim_0 \, x \ \ \Rightarrow \ \ \lim_{x\to0}\:(\sin x)^x=\lim_{x\to0}\:x^x.$$
