Chapter 3 - The Derivative - 3.2 Continuity - 3.2 Exercises - Page 147: 6

The function $f(x)$ is discontinuous at $x=0$ and $x=2$ (a) $f(0)$ does not exist. $f(2) $ does not exist. (b) $$ \lim _{x \rightarrow 0^{-}} f(x)=- \infty \quad \quad \text {(limit does not exist) } $$ and $$ \lim _{x \rightarrow 2^{-}} f(x)=-2 $$ (c) $$ \lim _{x \rightarrow 0^{+}} f(x)=-\infty \quad \quad \text {(limit does not exist) } $$ and $$ \lim _{x \rightarrow 2^{+}} f(x)=-2 $$ (d) Since $$ \lim _{x \rightarrow 0^{-}} f(x) \ne \lim _{x \rightarrow 0^{+}} f(x) $$ then, $$ \lim _{x \rightarrow 0} f(x) \quad \quad \text {(does not exist) } $$ and $$ \lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)=-2 $$ then, $$ \lim _{x \rightarrow 2} f(x)=-2. $$ (e) $f(0)$ does not exist. $$ \lim _{x \rightarrow 0} f(x) \quad \quad \text {(does not exist) } $$ $f(2) $ does not exist.

The function $f(x)$ is discontinuous at $x=0$ and $x=2$ (a) $f(0)$ does not exist. $f(2) $ does not exist. (b) $$ \lim _{x \rightarrow 0^{-}} f(x)=- \infty \quad \quad \text {(limit does not exist) } $$ and $$ \lim _{x \rightarrow 2^{-}} f(x)=-2 $$ (c) $$ \lim _{x \rightarrow 0^{+}} f(x)=-\infty \quad \quad \text {(limit does not exist) } $$ and $$ \lim _{x \rightarrow 2^{+}} f(x)=-2 $$ (d) Since $$ \lim _{x \rightarrow 0^{-}} f(x) \ne \lim _{x \rightarrow 0^{+}} f(x) $$ then, $$ \lim _{x \rightarrow 0} f(x) \quad \quad \text {(does not exist) } $$ and $$ \lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)=-2 $$ then, $$ \lim _{x \rightarrow 2} f(x)=-2. $$ (e) $f(0)$ does not exist. $$ \lim _{x \rightarrow 0} f(x) \quad \quad \text {(does not exist) } $$ $f(2) $ does not exist.