Question

find each indicated quantity if it exists. let f(x) = {x^2, for x < - 2; 2x, for x > - 2}. complete parts (a) through (d). (a) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim_{x→ - 2^+} f(x) = - 4 (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. lim_{x→ - 2^-} f(x) = (type an integer.) b. the limit does not exist.

Explanation:

Step1: Recall right - hand limit definition

For $\lim_{x
ightarrow - 2^{+}}f(x)$, we use the part of the function where $x > - 2$. Since $f(x)=2x$ for $x > - 2$, we substitute $x=-2$ into $2x$.

Step2: Calculate the right - hand limit

Substitute $x = - 2$ into $y = 2x$. We get $y=2\times(-2)=-4$. So $\lim_{x
ightarrow - 2^{+}}f(x)=-4$.

Step3: Recall left - hand limit definition

For $\lim_{x
ightarrow - 2^{-}}f(x)$, we use the part of the function where $x < - 2$. Since $f(x)=x^{2}$ for $x < - 2$, we substitute $x=-2$ into $x^{2}$.

Step4: Calculate the left - hand limit

Substitute $x=-2$ into $y = x^{2}$. We get $y=(-2)^{2}=4$. So $\lim_{x
ightarrow - 2^{-}}f(x)=4$.

Answer:

(A) A. $\lim_{x
ightarrow - 2^{+}}f(x)=-4$
(B) A. $\lim_{x
ightarrow - 2^{-}}f(x)=4$