Find limit of the quotient Suppose $f,g$ are functions on $I=(0,2)$ such that $$\lim_{x \to 1} f(x) = 1$$ and $$\lim_{x \to 1} g(x)= -\infty.$$ Prove that $$\lim_{x \to 1} f(x)/g(x) = 0.$$
for this question the main idea that i have going on is that ima have $|f(x)-1|< \epsilon$   and i have to somehow show that $|f(x)/g(x) - 0| < \epsilon$ but that's as far as i have gone.

Let $\epsilon > 0$.
Then there is some $\delta > 0$ such that
$|f(x) - 1| < \epsilon$
for all $0 < |x-1| < \delta$ and there is some $\delta' > 0$
such that
$|g(x)| > 1/\epsilon$ for all $0 < |x-1| < \delta'$.
Thus for all $0 < |x-1| < \min \{\delta, \delta'\}$ we have
$$\bigg| \frac{f(x)}{g(x)} \bigg| < \frac{1+\epsilon}{1/\epsilon} = \epsilon + \epsilon^{2}.$$
