TITLE: On 'sum of limits is the limit of the sum'
QUESTION [0 upvotes]: I am not sure about what happens with this property when the limits are not real numbers, i.e., they exist, but are $\infty$. Say $\lim_{x \to a} f(x) = \infty$ and $\lim_{x \to a} g(x) = -\infty$. In this case, can we apply the property? Is it true that $\lim_{x \to a} f(x) + \lim_{x \to a} g(x) = \lim_{x \to a} \left( f(x)+g(x) \right)$? If it is not, is there any way to deal with such a sum of limits? I'll leave an example from where I asked myself about this.
$$\lim_{x \to 0} \left( \frac{\pi}{2} - x \right) \ln^2 {x} - \frac{\pi}{2} \lim_{x \to 0} \frac{1}{x^2}$$
I think the result would be $- \infty$ if I could apply this property and bring them into a single limit, but I wasn't sure about that as they both are $\infty$ so that's why I'm asking.

REPLY [1 votes]: If$$\lim_{x\to a}f(x)=\infty\quad\text{and}\quad\lim_{x\to a}g(x)=-\infty,$$then $\displaystyle\lim_{x\to a}\bigl(f(x)+g(x)\bigr)$ can be anything. For instance, if $b\in\Bbb R$, if $f(x)=\frac1{(x-a)^2}+b$ and if $g(x)=-\frac1{(x-a)^2}$, then$$\lim_{x\to a}f(x)=\infty,\ \lim_{x\to a}g(x)=-\infty\text{ and }\lim_{x\to a}\bigl(f(x)+g(x)\bigr)=b.$$
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