Question

find each indicated quantity if it exists. let f(x) = {x², 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)= 4 (type an integer.) b. the limit does not exist. (c) 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: Find right - hand limit

For $x\to - 2^{+}$, we use $f(x)=2x$. Substitute $x = - 2$ into $2x$. So, $\lim_{x\to - 2^{+}}f(x)=2\times(-2)=-4$.

Step2: Find left - hand limit

For $x\to - 2^{-}$, we use $f(x)=x^{2}$. Substitute $x=-2$ into $x^{2}$. So, $\lim_{x\to - 2^{-}}f(x)=(-2)^{2}=4$.

Step3: Check overall limit

Since $\lim_{x\to - 2^{-}}f(x)=4$ and $\lim_{x\to - 2^{+}}f(x)=-4$, and $4
eq - 4$, the overall limit $\lim_{x\to - 2}f(x)$ does not exist.

Answer:

C. B. The limit does not exist.