Calculate the limit: $\lim_{x\rightarrow 0}\frac{a^{x}-b^{x}}{x}$ 
Calculate the limit: $$\lim_{x\rightarrow 0}\frac{a^{x}-b^{x}}{x} \ \ \ \ a,b > 0$$

If I try calculating the limit like that, I will get $\frac{1-1}{0}=\frac{0}{0}$, so I will have to use L'Hôpital:
$$\lim_{x\rightarrow 0} \frac{\ln a\cdot a^{x}- \ln b \cdot b^{x}}{1}=\lim_{x\rightarrow 0}\left(\ln a \cdot a^{x}- \ln b \cdot b^{x}\right)= \ln a - \ln b$$
So this will be the limit then. But I do correctly? Let's say it's alright, what if professor asked me if this is convergent? I don't know because $a,b > 0$ and so they could be $\infty$ too, no? So I cannot really say if it converges or diverges?

It's correct. Moreover, $a$ and $b$ can't be $\infty$. Since the limit is finite, it converges. 
