Question

the graph below is the function $f(x)$
find $\lim\limits_{x \to 2^{-}} f(x)$
find $\lim\limits_{x \to 2^{+}} f(x)$
find $\lim\limits_{x \to 2} f(x)$
find $f(2)$
question help: video
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Explanation:

Step1: Analyze left - hand limit as \(x

ightarrow2^{-}\)
To find \(\lim_{x
ightarrow2^{-}}f(x)\), we look at the values of \(f(x)\) as \(x\) approaches \(2\) from the left (values less than \(2\)). From the graph, as \(x\) gets closer to \(2\) from the left side, the function values approach \(- 1\). So \(\lim_{x
ightarrow2^{-}}f(x)=-1\).

Step2: Analyze right - hand limit as \(x

ightarrow2^{+}\)
To find \(\lim_{x
ightarrow2^{+}}f(x)\), we look at the values of \(f(x)\) as \(x\) approaches \(2\) from the right (values greater than \(2\)). From the graph, as \(x\) gets closer to \(2\) from the right side, the function values approach \(-1\). So \(\lim_{x
ightarrow2^{+}}f(x)=-1\).

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

ightarrow2\)
The two - sided limit \(\lim_{x
ightarrow2}f(x)\) exists if the left - hand limit and the right - hand limit are equal. Since \(\lim_{x
ightarrow2^{-}}f(x)=\lim_{x
ightarrow2^{+}}f(x)=-1\), then \(\lim_{x
ightarrow2}f(x)=-1\).

Step4: Find \(f(2)\)

To find \(f(2)\), we look at the point on the graph where \(x = 2\). The filled - in dot (the actual value of the function at \(x = 2\)) is at \(y = 3\). So \(f(2)=3\).

Answer:

s:
\(\lim_{x
ightarrow2^{-}}f(x)=\boldsymbol{-1}\)

\(\lim_{x
ightarrow2^{+}}f(x)=\boldsymbol{-1}\)

\(\lim_{x
ightarrow2}f(x)=\boldsymbol{-1}\)

\(f(2)=\boldsymbol{3}\)