Question

the graph of the function ( h ) is shown below, along with its asymptote.
find the following limits.
if necessary, select the most informative answer from ( infty ), ( -infty ), and \does not exist\.
( limlimits_{x \to 1^-} h(x) = )
( limlimits_{x \to 1^+} h(x) = )
( limlimits_{x \to 1} h(x) = )

Explanation:

Step1: Analyze left - hand limit as \(x

ightarrow1^{-}\)
To find \(\lim_{x
ightarrow1^{-}}h(x)\), we look at the behavior of the function \(h(x)\) as \(x\) approaches \(1\) from the left - hand side (values of \(x\) that are less than \(1\) and getting closer to \(1\)). From the graph, as \(x\) approaches \(1\) from the left, the function values are going downwards without bound. So, \(\lim_{x
ightarrow1^{-}}h(x)=-\infty\).

Step2: Analyze right - hand limit as \(x

ightarrow1^{+}\)
To find \(\lim_{x
ightarrow1^{+}}h(x)\), we look at the behavior of the function \(h(x)\) as \(x\) approaches \(1\) from the right - hand side (values of \(x\) that are greater than \(1\) and getting closer to \(1\)). From the graph, as \(x\) approaches \(1\) from the right, the function values are going upwards without bound. So, \(\lim_{x
ightarrow1^{+}}h(x)=\infty\).

Step3: Analyze the two - sided limit as \(x

ightarrow1\)
For the two - sided limit \(\lim_{x
ightarrow1}h(x)\) to exist, the left - hand limit and the right - hand limit must be equal. Since \(\lim_{x
ightarrow1^{-}}h(x)=-\infty\) and \(\lim_{x
ightarrow1^{+}}h(x)=\infty\), and \(-\infty
eq\infty\), the two - sided limit \(\lim_{x
ightarrow1}h(x)\) does not exist.

Answer:

\(\lim_{x
ightarrow1^{-}}h(x)=-\infty\)

\(\lim_{x
ightarrow1^{+}}h(x)=\infty\)

\(\lim_{x
ightarrow1}h(x)=\text{Does not exist}\)