Question

determine whether lim f(x) exists. if so, give the limit. x→3 select the correct choice below and, if necessary, fill in any answer boxes in your choice. a. lim f(x)= (type an integer or a decimal.) x→3 b. the limit does not exist.

Explanation:

Step1: Recall limit - existence condition

The limit $\lim_{x
ightarrow a}f(x)$ exists if and only if $\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)$. We need to find the left - hand limit and the right - hand limit as $x
ightarrow3$.

Step2: Find left - hand limit

As $x$ approaches $3$ from the left ($x
ightarrow3^{-}$), we follow the part of the graph to the left of $x = 3$. The $y$ - value approaches $4$. So, $\lim_{x
ightarrow3^{-}}f(x)=4$.

Step3: Find right - hand limit

As $x$ approaches $3$ from the right ($x
ightarrow3^{+}$), we follow the part of the graph to the right of $x = 3$. The $y$ - value approaches $2$. So, $\lim_{x
ightarrow3^{+}}f(x)=2$.

Step4: Check limit existence

Since $\lim_{x
ightarrow3^{-}}f(x)=4$ and $\lim_{x
ightarrow3^{+}}f(x)=2$, and $4
eq2$, the limit $\lim_{x
ightarrow3}f(x)$ does not exist.

Answer:

B. The limit does not exist.