#### Answer

(a) True (b) False (c) False (d) True (e) True (f) True
(g) True (h) True (i) True (j) False (k) True

#### Work Step by Step

(a) $\lim_{x\to-1^+}f(x)=1$
TRUE. We do see intuitively that as $x$ approaches $-1$ from the right, $f(x)$ gets arbitrarily close to $1$.
(b) $\lim_{x\to2}f(x)$ does not exist
FALSE. We can see intuitively:
- $\lim_{x\to2^-}f(x)=1$, because as $x$ approaches $2$ from the left, $f(x)$ gets arbitrarily close to $1$.
- $\lim_{x\to2^+}f(x)=1$, because as $x$ approaches $2$ from the right, $f(x)$ gets arbitrarily close to $1$.
Therefore, because $\lim_{x\to2^+}f(x)=\lim_{x\to2^-}f(x)$, so $\lim_{x\to2}f(x)$ does exist and equals $1$.
(c) $\lim_{x\to2}f(x)=2$
FALSE. We showed in (b) that $\lim_{x\to2}f(x)=1$.
(d) $\lim_{x\to1^-}f(x)=2$
TRUE. We do see intuitively that as $x$ approaches $1$ from the left, $f(x)$ gets arbitrarily close to $2$.
(e) $\lim_{x\to1^+}f(x)=1$
TRUE. We do see intuitively that as $x$ approaches $1$ from the right, $f(x)$ gets arbitrarily close to $1$.
(f) $\lim_{x\to1}f(x)$ does not exist.
TRUE. Since $\lim_{x\to1^-}f(x)\ne\lim_{x\to1^+}f(x)$, $\lim_{x\to1}f(x)$ does not exist.
(g) $\lim_{x\to0^+}f(x)=\lim_{x\to0^-}f(x)$
TRUE. We do see intuitively that as $x$ approaches $0$ from both the left and the right, $f(x)$ gets arbitrarily close to $0$. So $\lim_{x\to0^+}f(x)=\lim_{x\to0^-}f(x)=0$
(h) $\lim_{x\to c}f(x)$ exists at every $c$ in the open interval $(-1,1)$
TRUE. At all $c$ in the open interval $(-1,1)$, as $x$ approaches $c$ from both the left and the right, $f(x)$ gets arbitrarily close to one value of $f(x)$ only, meaning $\lim_{x\to c}f(x)$ exists.
(i) $\lim_{x\to c}f(x)$ exists at every $c$ in the open interval $(1,3)$
TRUE. At all $c$ in the open interval $(1,3)$, as $x$ approaches $c$ from both the left and the right, $f(x)$ gets arbitrarily close to one value of $f(x)$ only, meaning $\lim_{x\to c}f(x)$ exists.
(j) $\lim_{x\to-1^-}f(x)=0$
FALSE. There are no values of $f(x)$ as $x$ approaches $-1$ from the left. In other words, $\lim_{x\to-1^-}f(x)$ does not exist.
(k) $\lim_{x\to3^+}f(x)$ does not exist.
TRUE. There are no values of $f(x)$ as $x$ approaches $3$ from the right. In other words, $\lim_{x\to3^+}f(x)$ does not exist.