Question

use the graph to evaluate the limit.
lim f(x)
x→0
∞
-1
does not exist
1

Explanation:

Step1: Recall limit - definition

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 $x
ightarrow0$.

Step2: Find left - hand limit

As $x$ approaches $0$ from the left side ($x
ightarrow0^{-}$), we look at the values of $y = f(x)$ for $x\lt0$. From the graph, as $x$ approaches $0$ from the left, $y=-1$. So, $\lim_{x
ightarrow0^{-}}f(x)=-1$.

Step3: Find right - hand limit

As $x$ approaches $0$ from the right side ($x
ightarrow0^{+}$), we look at the values of $y = f(x)$ for $x\gt0$. From the graph, as $x$ approaches $0$ from the right, $y = 1$. So, $\lim_{x
ightarrow0^{+}}f(x)=1$.

Step4: Compare left and right limits

Since $\lim_{x
ightarrow0^{-}}f(x)=-1$ and $\lim_{x
ightarrow0^{+}}f(x)=1$, and $-1
eq1$.

Answer:

does not exist