Question

use the graph of the function f shown to estimate the following limits and the function value. complete parts (a) through (d). (a) find lim f(x). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim f(x)=□ (type an integer.) b. the limit does not exist. (b) find lim f(x). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim f(x)=□ (type an integer.) b. the limit does not exist. (c) find lim f(x). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim f(x)=□ (type an integer.) b. the limit does not exist. (d) find the function value f(2). 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: Recall limit - from - left concept

To find $\lim_{x
ightarrow2^{-}}f(x)$, we look at the values of the function $f(x)$ as $x$ approaches $2$ from the left - hand side. Looking at the graph, as $x$ gets closer and closer to $2$ from the left, $f(x)$ approaches $2$.

Step2: Recall limit - from - right concept

To find $\lim_{x
ightarrow2^{+}}f(x)$, we look at the values of the function $f(x)$ as $x$ approaches $2$ from the right - hand side. From the graph, as $x$ gets closer and closer to $2$ from the right, $f(x)$ approaches $3$.

Step3: Recall two - sided limit concept

The two - sided limit $\lim_{x
ightarrow2}f(x)$ exists if and only if $\lim_{x
ightarrow2^{-}}f(x)=\lim_{x
ightarrow2^{+}}f(x)$. Since $\lim_{x
ightarrow2^{-}}f(x) = 2$ and $\lim_{x
ightarrow2^{+}}f(x)=3$, $\lim_{x
ightarrow2}f(x)$ does not exist.

Step4: Recall function - value concept

To find $f(2)$, we look at the value of the function at $x = 2$. If there is a closed - circle at $x = 2$ on the graph, that is the function value. If there is no closed - circle, the function is not defined at that point. From the graph, the function is not defined at $x = 2$.

Answer:

(A) A. $\lim_{x
ightarrow2^{-}}f(x)=2$
(B) A. $\lim_{x
ightarrow2^{+}}f(x)=3$
(C) B. The limit does not exist.
(D) B. The function is not defined at $x = 2$.