Can someone please explain to me how to prove the following:
If $\displaystyle\lim_{x\to c}F(x) = L $ and if $\displaystyle\lim_{x\to c}H(x) = H$ such that $H(x)\ne0$ for all $x$ in the domain, then $\displaystyle\lim_{x\to c}F(x)/H(x) = L/H$.
I dont want the whole proof, just give me some hints so that I can work on the proof on my own.
Thanks!
Note that $$\frac{F(x)}{H(x)} - \frac{L}{H} = \frac{F(x)H - L H(x)}{H(x)H} = \frac{(F(x)H - LH ) +(LH- L H(X))}{H(x)H}.$$
Then show that the two summands in the numerator become small, while the denominator stays away from $0$ so that the quotient in total is small.
(Note: I assume the limit $H$ is also assumed non-zero. This is not explicitly stated and not a consequence of the fact that $H(x)$ is non-zero.)