use the graph to answer these questions\na find $lim_{x\rightarrow4^{+}}f(x)$ and $lim_{x\rightarrow4^{-}}f(x…

use the graph to answer these questions\na find $lim_{x\rightarrow4^{+}}f(x)$ and $lim_{x\rightarrow4^{-}}f(x)$\nb find $lim_{x\rightarrow4}f(x)$\nc find $f(4)$\nd is $f(x)$ continuous at $x = 4$? why or why not?\na. select the correct choice below and, if necessary, fill in the answer box to complete your choice\n○a. $lim_{x\rightarrow4^{+}}f(x)=$ (simplify your answer.)\n○b. the limit does not exist\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice\n○a. $lim_{x\rightarrow4^{-}}f(x)=$ (simplify your answer.)\n○b. the limit does not exist

use the graph to answer these questions\na find $lim_{x\rightarrow4^{+}}f(x)$ and $lim_{x\rightarrow4^{-}}f(x)$\nb find $lim_{x\rightarrow4}f(x)$\nc find $f(4)$\nd is $f(x)$ continuous at $x = 4$? why or why not?\na. select the correct choice below and, if necessary, fill in the answer box to complete your choice\n○a. $lim_{x\rightarrow4^{+}}f(x)=$ (simplify your answer.)\n○b. the limit does not exist\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice\n○a. $lim_{x\rightarrow4^{-}}f(x)=$ (simplify your answer.)\n○b. the limit does not exist

Answer

Explanation:

Step1: Analyze right - hand limit

As (x\to4^{+}), we look at the part of the graph for (x > 4). Following the curve for (x>4) as (x) approaches 4 from the right, we can see that the (y) - values approach 5. So, (\lim_{x\to4^{+}}f(x)=5).

Step2: Analyze left - hand limit

As (x\to4^{-}), we look at the part of the graph for (x < 4). Following the curve for (x < 4) as (x) approaches 4 from the left, we can see that the (y) - values approach 2. So, (\lim_{x\to4^{-}}f(x)=2).

Step3: Analyze overall limit

The overall limit (\lim_{x\to4}f(x)) exists if and only if (\lim_{x\to4^{+}}f(x)=\lim_{x\to4^{-}}f(x)). Since (\lim_{x\to4^{+}}f(x) = 5) and (\lim_{x\to4^{-}}f(x)=2), (\lim_{x\to4}f(x)) does not exist.

Step4: Find (f(4))

The solid dot on the graph at (x = 4) has a (y) - value of 2. So, (f(4)=2).

Step5: Check continuity

A function (y = f(x)) is continuous at (x=a) if (\lim_{x\to a}f(x)=f(a)). Here, (\lim_{x\to4}f(x)) does not exist (because (\lim_{x\to4^{+}}f(x)\neq\lim_{x\to4^{-}}f(x))) and (f(4) = 2). So, (f(x)) is not continuous at (x = 4).

Answer:

a. (\lim_{x\to4^{+}}f(x)=5), (\lim_{x\to4^{-}}f(x)=2) b. The limit does not exist c. (f(4)=2) d. (f(x)) is not continuous at (x = 4) because (\lim_{x\to4}f(x)) does not exist.