Answer
(a) $2$ (b) $1$ (c) Does not exist. (d) $4$ (e) $4$ (f) $4$ (g) $0$ (h) $5$
Work Step by Step
Based on the given piece-wise function, we have:
(a) For $x\to -1^-$, we use the first function: $\lim_{x\to -1^-}f(x)=2$
(b) For $x\to -1^+$, we use the second function: $\lim_{x\to -1^+}f(x)=(-1)^2=1$
(c) As $\lim_{x\to -1^-}f(x)\ne \lim_{x\to -1^+}f(x)=(-1)^2$, the limit $\lim_{x\to -1}f(x)$ does not exist
(d) For $x\to 2^-$, we use the second function: $\lim_{x\to 2^-}f(x)=(2)^2=4$
(e) For $x\to 2^+$, we use the third function: $\lim_{x\to 2^+}f(x)=2+2=4$
(f) Based on the above resutls, $\lim_{x\to 2}f(x)=4$
(g) For $x\to 0$, we use the second function: $\lim_{x\to 0}f(x)=(0)^2=0$
(h) For $x\to 3$, we use the third function: $\lim_{x\to 3}(f(x)^2=3+2=5$
