Limits approaching from both sides go to infinity

Question

Suppose that $\lim_{x \to a} f(x) = \infty$. Prove that we then have $\lim_{x \to a^+} f(x) = \infty$ and $\lim_{x \to a^-} f(x) = \infty$ from the definitions using epsilon-delta methods.

Answer 1

Hint: Suppose $\lim_{x \to a} f(x) = \infty$. So for all $M > 0$, there is a $\delta > 0$ such that $|x - a|<\delta$ implies $f(x) > M$.

Now, look at $|x-a|<\delta$. In other words, try to work with $-\delta < x-a < \delta$.

It may help to recall the definitions of $\lim_{x \to a^-} f(x) = \infty$ and $\lim_{x \to a^+} f(x) = \infty$. Show how you will get to those definitions, to establish the fact that $\lim_{x \to a^-} f(x) = \infty$ and $\lim_{x \to a^+} f(x) = \infty$.