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. lim f(x) = 4 (type an integer.) x→ - 2⁻ 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 f(x) = (type an integer.) x→ - 2 b. the limit does not exist. (d) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. f(-2)= (type an integer.) b. the function is not defined at x = - 2.

Explanation:

Step1: Calculate left - hand limit

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

Step2: Calculate right - hand limit

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

Step3: Determine the two - sided limit

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

Step4: Check the function value at $x=-2$

The function $f(x)=

$$\begin{cases}x^{2},&x < - 2\\2x,&x > - 2\end{cases}$$

$ is not defined at $x = - 2$.

Answer:

(A) $\lim_{x\to - 2^{-}}f(x)=4$
(C) B. The limit does not exist.
(D) B. The function is not defined at $x=-2$.