[Math] How to find the limit, without L’Hôpital’s rule, of a function with exponentials

Question

I have done some digging and I cannot find any posts addressing limits with exponentials and without L'Hôpital's rule.

I have one of these questions for my assignment, but for ethical reasons I have made up a similar function:

Find the following limit without L'Hôpital's rule:
$$\lim_{x\to0}\frac{2^x-7^x}{2x}$$

Answer 1

$$ \lim_{x\to 0}\frac{2^x-7^x}{2x} = \frac12\lim_{x\to 0}\frac{2^x-7^x-0}{x-0}$$ and the second limit is by definition the derivative of $x\mapsto 2^x-7^x$ at $x=0$. Differentiate this function symbolically and you're done.