Sum rule in limits

Question

Suppose we want to find $\lim_{x \to a}(f(x) - g(x))$ . We are not aware about the existence of $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ . Can we use sum rule and rewrite it to $\lim_{x \to a}f(x) - \lim_{x \to a}g(x)$ ? For example if after rewriting it , we found that $\lim_{x \to a} f(x) = \infty $ and $\lim_{x \to a} g(x) = l$ then this conclusion $\lim_{x \to a}(f(x) - g(x)) = \infty$ is right ? I always have trouble in the using sum and multiplication rules in order to computing limits .

Answer 1

Just if there is $\lim\limits_{x\rightarrow a}f(x)=A$ and there is $\lim\limits_{x\rightarrow a}g(x)=B$ then there is $\lim\limits_{x\rightarrow a}(f(x)+g(x))=A+B$

About the second question.

If there is $\lim\limits_{x\rightarrow a}f(x)=A$ and $\lim\limits_{x\rightarrow a}g(x)=\infty$ then $\lim\limits_{x\rightarrow a}(f(x)+g(x))=\infty.$