Inverting a fraction in a limit

Question

I've seen in a limit problem(limit of a fraction) that the fraction can be inverted and then proceed on the new fraction. Is this ok? Can someone show a proof?

Answer 1

If you have: $$\lim_{x \to a} f(x) = L$$ with $L \ne 0$, then you also have: $$\lim_{x \to a} \frac{1}{f(x)} = \frac{1}{L}$$ So yes: you can consider the inverted fraction, but then you get the reciprocal of the (non-zero) limit as well.