use the graph to answer these questions\na find lim f(x) and lim f(x)\nx→3+ x→3-\nb find lim f(x)\nx→3\nc…

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.