Question

the function h is graphed below.
find the following limits. if a limit does not exist, click on \does not exist\.
$$\lim_{x \to 3^{-}} h(x) = \square$$
$$\lim_{x \to 3^{+}} h(x) = \square$$
$$\lim_{x \to 3} h(x) = \square$$

Explanation:

Step1: Analyze the left - hand limit ($\lim_{x

ightarrow3^{-}}h(x)$)
As \(x\) approaches \(3\) from the left (\(x
ightarrow3^{-}\)), we look at the behavior of the function values. From the graph, as \(x\) gets closer and closer to \(3\) from the left - hand side, the \(y\) - values of the function \(h(x)\) approach \(0\). So, \(\lim_{x
ightarrow3^{-}}h(x)=0\).

Step2: Analyze the right - hand limit ($\lim_{x

ightarrow3^{+}}h(x)$)
As \(x\) approaches \(3\) from the right (\(x
ightarrow3^{+}\)), we observe the behavior of the function. From the graph, as \(x\) gets closer and closer to \(3\) from the right - hand side, the \(y\) - values of the function \(h(x)\) approach \(3\). So, \(\lim_{x
ightarrow3^{+}}h(x)=3\).

Step3: Analyze the two - sided limit ($\lim_{x

ightarrow3}h(x)$)
The two - sided limit \(\lim_{x
ightarrow a}f(x)\) exists if and only if \(\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)\). Here, since \(\lim_{x
ightarrow3^{-}}h(x) = 0\) and \(\lim_{x
ightarrow3^{+}}h(x)=3\), and \(0
eq3\), the two - sided limit \(\lim_{x
ightarrow3}h(x)\) does not exist.

Answer:

\(\lim_{x
ightarrow3^{-}}h(x)=0\), \(\lim_{x
ightarrow3^{+}}h(x)=3\), \(\lim_{x
ightarrow3}h(x)\) does not exist.