calculus - Evaluating $lim_{xrightarrow 0} frac{ln(1-x)+sin(x)}{x^2 e^x} $

I got this exercise which I quite frankly can't wrap my head around

$$\lim_{x\rightarrow 0} \frac{\ln(1-x)+\sin(x)}{x^2 e^x} $$

The result should be:$\ -\frac{1}{2} $

I tried by derivating the whole function and that led me nowhere. If I put in $\ x = 0 $, I get $\ \frac{0}{0} $ which means I can apply L'Hôpital's rule but that led me to the wrong answer as well. Obviously I'm doing something wrong. I had a look at Wolfram Alphas step-by-step solution to this exercise but I didn't really understand it.

I'm appreciative with any possible help.

Thanks,

Michael.

Answer

The $e^x$ part safely has limit $1$, so we can forget about it and calculate
$$\lim_{x\to 0}\frac{\ln(1-x)+\sin x}{x^2}.$$
Now L'Hospital's Rule works nicely, with smaller probability of error than if we keep the $e^x$ part.

I prefer a somewhat more "hands on" approach that uses the first few terms of the power series expansions of $\ln(1-x)$ and $\sin x$. Note that $\ln(1-x)=-x-\dfrac{x^2}{2}+o(x^2)$ while $\sin x=x+o(x^2)$. Add. We get $-\dfrac{x^2}{2}+o(x^2)$. Divide by $x^2$. So our expression is $-\dfrac{1}{2}+o(1)$, which is the desired result.