Answer
$a.\quad 2$
$b.\quad 0$
$c.\quad $does not exist.
Work Step by Step
a.
As x approaches -1 from the left, $f(x)=x^{2}+1$, a polynomial, so
$\displaystyle \lim_{x\rightarrow-1^{-}}f(x)=\lim_{x\rightarrow-1^{-}}\left(x^{2}+1\right)=(-1)^{2}+1=2$
b.
As x approaches -1 from the right, $f(x)=\sqrt{x+1}$,
$\displaystyle \lim_{x\rightarrow-1^{+}}f(x)=\lim_{x\rightarrow-1^{+}}\sqrt{x+1}$
... apply the fractional power rule (Th.2.3.7)
$=\sqrt{\lim_{x\rightarrow-1^{+}}(x+1)}$
the limit of a linear function = f(-1)
$=\sqrt{-1+1}=0$
c.
The one-sided limits exist, but they are not equal.
$\displaystyle \lim_{x\rightarrow-1}f(x)$ does not exist.
