On adding terms to limits Is it always possible to add terms into limits, like in the following example? (Or must certain conditions be fulfilled first, such as for example the numerator by itself must converge etc)
$\lim_{h \to 0} {f(x)} = \lim_{h \to 0} \frac{e^xf(x)}{e^x}$

Suppose $f,g$ are defined on the punctured neighbourhood of point $a$ and limits $$\lim_{x \to a} {f(x)},\quad \lim_{x \to a} {g(x)}$$exist.
Then $\lim_{x \to a} {f(x)\cdot g(x)}$ exists and
$$\lim_{x \to a} {f(x)\cdot g(x)}=\lim_{x \to a} {f(x)}\cdot \lim_{x \to a} {g(x)}$$
