Question

use the graph to answer these questions
a find lim f(x) and lim f(x)
x→3+ x→3-
b find lim f(x)
x→3
c find f(3)
d is f(x) continuous at x = 3? why or why not?

Explanation:

Step1: Find right - hand limit

As \(x\to3^{+}\), we look at the values of \(y = f(x)\) as \(x\) approaches 3 from the right side of the graph. Following the curve for \(x>3\), we see that \(\lim_{x\to3^{+}}f(x)= 4\).

Step2: Find left - hand limit

As \(x\to3^{-}\), we look at the values of \(y = f(x)\) as \(x\) approaches 3 from the left side of the graph. Following the curve for \(x < 3\), we see that \(\lim_{x\to3^{-}}f(x)=2\).

Step3: Find the overall limit

The overall limit \(\lim_{x\to3}f(x)\) exists if \(\lim_{x\to3^{+}}f(x)=\lim_{x\to3^{-}}f(x)\). Since \(\lim_{x\to3^{+}}f(x) = 4\) and \(\lim_{x\to3^{-}}f(x)=2\), \(\lim_{x\to3}f(x)\) does not exist.

Step4: Find \(f(3)\)

Looking at the graph, when \(x = 3\), the filled - in point has a \(y\) - value of 4. So \(f(3)=4\).

Step5: Check continuity

A function \(y = f(x)\) is continuous at \(x=a\) if \(\lim_{x\to a}f(x)=f(a)\). Here, \(\lim_{x\to3}f(x)\) does not exist and \(f(3) = 4\). So \(f(x)\) is not continuous at \(x = 3\) because \(\lim_{x\to3}f(x)\) does not exist.

Answer:

a. \(\lim_{x\to3^{+}}f(x)=4\), \(\lim_{x\to3^{-}}f(x)=2\)
b. \(\lim_{x\to3}f(x)\) does not exist
c. \(f(3)=4\)
d. \(f(x)\) is not continuous at \(x = 3\) because \(\lim_{x\to3}f(x)\) does not exist.