Chapter 1.7, Problem 44E

Suppose that ${\lim }_{x\to a}f\left(x\right)=\infty$ and ${\lim }_{x\to a}g\left(x\right)=c$, where c is a real number. Prove each statement.

(a) $\underset{x\to a}{\lim }[f\left(x\right)+g\left(x\right)]=\infty$

(b) $\underset{x\to a}{\lim }[f\left(x\right)g\left(x\right)]=\infty \text{ if }c>0$

(c) $\underset{x\to a}{\lim }[f\left(x\right)g\left(x\right)]=-\infty \text{ if }c<0$