Chapter 2.4, Problem 44E

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

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

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

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