Chapter 2.3, Problem 71E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.

1. a. If $\underset{x\to a}{\lim }f(x)=L$, then f(a) = L.
2. b. If $\underset{x\to a^{-}}{\lim }f(x)=L$, then $\underset{x\to a^{+}}{\lim }f(x)=L$.
3. c. If $\underset{x\to a}{\lim }f(x)=L$ and $\underset{x\to a}{\lim }g(x)=L$, then f(a) = g(a).
4. d. The limit $\underset{x\to a}{\lim }\frac{f(x)}{g(x)}$ does not exist if g(a) = 0.
5. e. If $\underset{x\to 1^{+}}{\lim }\sqrt{f(x)}=\sqrt{\underset{x\to 1^{+}}{\lim }}f(x)$, it follows that $\underset{x\to 1}{\lim }\sqrt{f(x)}=\sqrt{\underset{x\to 1}{\lim }f(x)}$.