Question

use the graph to answer these questions
a find $lim_{x
ightarrow4^{+}}f(x)$ and $lim_{x
ightarrow4^{-}}f(x)$
b find $lim_{x
ightarrow4}f(x)$
c find $f(4)$
d is $f(x)$ continuous at $x = 4$? why or why not?
a. select the correct choice below and, if necessary, fill in the answer box to complete your choice
○a. $lim_{x
ightarrow4^{+}}f(x)=$ (simplify your answer.)
○b. the limit does not exist
select the correct choice below and, if necessary, fill in the answer box to complete your choice
○a. $lim_{x
ightarrow4^{-}}f(x)=$ (simplify your answer.)
○b. the limit does not exist

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 \(\lim_{x\to4^{+}}f(x)
eq\lim_{x\to4^{-}}f(x)\)) 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.