Kind of limit I don't know how to solve

Question

Suppose that we have three functions with the following behavior:

$$\lim_{x\to x_0} f(x)=0$$

$$\lim_{x\to x_0} g(x)=0$$

$$\lim_{x\to x_0} h(x)=\infty$$

I don't know how to approach a limit of the form:

$$\lim_{x\to x_0} \frac {f(x)h(x)}{g(x)}$$

Is there a general way to threat them ? Every manipulation I try to do with de l'Hopital doesn't lead me very further, should I try Taylor ?

I give an explicit example of what I'm talking about:

$$\lim_{x\to 0} \frac {\Gamma (x)\sin (x)}{e^x-1}$$

Answer 1

In general, Taylor/Laurent series may be a good idea. However, in your example, we have that $\lim_{x\to0}\frac{\sin x}{e^x-1}=1$ (for example, by l'Hopital) so that $$\lim_{x\to 0} \frac{\Gamma(x)\sin x}{e^x-1}=\lim_{x\to 0}\Gamma(x)$$

Answer 2

In this specific case, you should be able to determine that $\lim_{x \to 0}\frac{\sin x}{e^x - 1} = \lim_{x \to 0}\frac{\cos x}{e^x} = 1$.

Hence the overall limit is undefined.