the function, f(x), is plotted as shown. evaluate the following limits: lim x→2 f(x)= lim x→2 f(x)= lim x→2…

Explanation:

Step1: Recall left - hand limit definition

The left - hand limit $\lim_{x\rightarrow2^{-}}f(x)$ is the value the function approaches as $x$ approaches 2 from the left. Looking at the graph, as $x$ approaches 2 from the left, $y$ approaches - 3.

Step2: Recall right - hand limit definition

The right - hand limit $\lim_{x\rightarrow2^{+}}f(x)$ is the value the function approaches as $x$ approaches 2 from the right. From the graph, as $x$ approaches 2 from the right, $y$ approaches 3.

Step3: Recall two - sided limit condition

The two - sided limit $\lim_{x\rightarrow2}f(x)$ exists if and only if $\lim_{x\rightarrow2^{-}}f(x)=\lim_{x\rightarrow2^{+}}f(x)$. Since $\lim_{x\rightarrow2^{-}}f(x)= - 3$ and $\lim_{x\rightarrow2^{+}}f(x)=3$, $\lim_{x\rightarrow2}f(x)$ does not exist.

Answer:

$\lim_{x\rightarrow2^{-}}f(x)=-3$ $\lim_{x\rightarrow2^{+}}f(x)=3$ $\lim_{x\rightarrow2}f(x)$ does not exist