Question

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

Explanation:

Step1: Analyze left - hand limit as \(x

ightarrow1^{-}\)
To find \(\lim_{x
ightarrow1^{-}}g(x)\), we look at the behavior of the function \(g(x)\) as \(x\) approaches \(1\) from the left side (values of \(x\) less than \(1\)). From the graph, as \(x\) gets closer to \(1\) from the left, the function \(g(x)\) is decreasing without bound, which means it approaches \(-\infty\). So \(\lim_{x
ightarrow1^{-}}g(x)=-\infty\).

Step2: Analyze right - hand limit as \(x

ightarrow1^{+}\)
To find \(\lim_{x
ightarrow1^{+}}g(x)\), we look at the behavior of the function \(g(x)\) as \(x\) approaches \(1\) from the right side (values of \(x\) greater than \(1\)). From the graph, as \(x\) gets closer to \(1\) from the right, the function \(g(x)\) is increasing without bound, which means it approaches \(\infty\).

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

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

Answer:

\(\lim_{x
ightarrow1^{-}}g(x)=\boldsymbol{-\infty}\), \(\lim_{x
ightarrow1^{+}}g(x)=\boldsymbol{\infty}\), \(\lim_{x
ightarrow1}g(x)\) \(\boldsymbol{\text{Does Not Exist}}\)