limits of combined functions Suppose that $a \in \mathbb R$. If $\lim_{x \to a} f(x) = \infty$ and $\lim_{x \to a} g(x) = -\infty$, show that
$\lim_{x \to a} (f-g)(x) = \infty$.
So, here I guess we use the fact that if $\lim_{x \to a} f(x) = \ell$ and $\lim_{x \to a} g(x) = m$, Then $\lim_{x \to a} (f-g)(x) = \ell - m$
In this case, $\ell = \infty$ and $m = - \infty$. Therefore, $\infty + \infty = \infty$
Am I right?
 A: Intuitively, yes. But expressions like $\infty - (-\infty)$ tend to make mathematicians feel uneasy, and you are presumably taking real analysis. So let's be slightly more rigorous by using the limit definition.
Since $\lim_{x\to a} f(x) = \infty$ and $\lim_{x \to a} g(x) = -\infty$, we know that:


*

*$\forall B_1 > 0$, $\exists \delta_1 > 0$ such that $0 < |x - a| < \delta_1 \implies f(x) > B_1$

*$\forall B_2 > 0$, $\exists \delta_2 > 0$ such that $0 < |x - a| < \delta_2 \implies g(x) < -B_2$


Now given any $B > 0$, let $\delta = \min\{\delta_1, \delta_2\} > 0$, where $\delta_1$ and $\delta_2$ are the positive deltas that are guaranteed to exist if we take $B_1 = B/2 > 0$ and $B_2 = B/2 > 0$. Then if $0 < |x - a| < \delta$, it follows that:
\begin{align*}
f(x) - g(x)
&> \frac{B}{2} - g(x) &\text{since } 0 < |x - a| < \delta \leq \delta_1 \implies f(x) > \frac{B}{2} \\
&> \frac{B}{2} + \frac{B}{2} &\text{since } g(x) < -\frac{B}{2}\iff -g(x) > \frac{B}{2} \\
&= B
\end{align*}
as desired.
