Answer
a. True.
b. False.
c. True.
d. False.
Work Step by Step
a. True. Use the formula $\lim_{x\to c}(f(x)+g(x))=\lim_{x\to c}f(x)+\lim_{x\to c}g(x)$.
Since the second limit does not exist, the sum does not exist.
b. False. Let $g(x)=-f(x)$; it is possible that both $\lim_{x\to c}f(x)$ and $\lim_{x\to c}g(x)$ do not exist (such as $f(x)=1/x$) but their sum $\lim_{x\to c}(f(x)+g(x))=0$ does exist.
c. True. We know that $|f|$ is the composite of the absolute value function ($g(x)=|x|$) and the $f(x)$ function. Since both are continuous, then the composite will also be continuous.
d. False. Let $f(x)=\begin{cases} 1,\ x\gt 0 \\ -1,\ x\le0 \end{cases}$,
while $|f(x)|$ is continuous at $x=0$, $f(x)$ is not.
