Question

find each indicated quantity if it exists. let $f(x)=\begin{cases}x^{2},& \text{for }x < - 1\\-2x,& \text{for }x > - 1end{cases}$. complete parts (a) through (d). (a) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. $limlimits_{x
ightarrow - 1^{+}}f(x)=2$ (type an integer.) b. the limit does not exist. (b) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. $limlimits_{x
ightarrow - 1^{-}}f(x)=square$ (type an integer.) b. the limit does not exist.

Explanation:

Step1: Recall right - hand limit definition

For $\lim_{x
ightarrow - 1^{+}}f(x)$, since $x
ightarrow - 1^{+}$ means $x>-1$, we use $f(x)=-2x$.

Step2: Substitute $x = - 1$ into the function

Substitute $x=-1$ into $y = - 2x$. We get $y=-2\times(-1)=2$.

Step3: Recall left - hand limit definition

For $\lim_{x
ightarrow - 1^{-}}f(x)$, since $x
ightarrow - 1^{-}$ means $x < - 1$, we use $f(x)=x^{2}$.

Step4: Substitute $x=-1$ into the function

Substitute $x = - 1$ into $y=x^{2}$. We get $y=(-1)^{2}=1$.

Answer:

(A) A. $\lim_{x
ightarrow - 1^{+}}f(x)=2$
(B) A. $\lim_{x
ightarrow - 1^{-}}f(x)=1$