Q:

Need a hint evaluating $ \lim\limits_{x\to 0}\frac{x\ln{(\frac{\sin (x)}{x})}}{\sin (x) - x} $

I'm stuck with this. I've tried substituting $t$ for $\frac{\sin (x)}{x}$ and $\sin (x) - x$ but it doesn't work at all. 
A small hint would be greatly appreciated. 

A:

$$\lim_{x\to 0}\frac{x\ln{(\frac{\sin (x)}{x})}}{\sin (x) - x} =  \lim_{x\to 0}\frac{\ln{(1+\frac{\sin (x)-x}{x})}}{\frac{\sin (x) - x}{x}} =1 $$ Because $$\lim_{x\to0}\frac{\sin x-x}{x}=\lim_{x\to0}\frac{\sin x}{x}-1=1-1=0$$ and $$\lim_{t\to0}\frac{\ln(1+t)}{t}=1.$$

