Question

the graph below is the function $f(x)$. find $lim_{x
ightarrow - 1^{-}}f(x)=$ find $lim_{x
ightarrow - 1^{+}}f(x)=$ find $lim_{x
ightarrow - 1}f(x)=$ find $f(-1)=$

Explanation:

Step1: Analyze left - hand limit

As \(x\to - 1^{-}\), we look at the values of the function as \(x\) approaches \(-1\) from the left - hand side. Following the graph, as \(x\) approaches \(-1\) from the left, \(y = 1\). So, \(\lim_{x\to - 1^{-}}f(x)=1\).

Step2: Analyze right - hand limit

As \(x\to - 1^{+}\), we look at the values of the function as \(x\) approaches \(-1\) from the right - hand side. Following the graph, as \(x\) approaches \(-1\) from the right, \(y = 1\). So, \(\lim_{x\to - 1^{+}}f(x)=1\).

Step3: Analyze overall limit

Since \(\lim_{x\to - 1^{-}}f(x)=\lim_{x\to - 1^{+}}f(x) = 1\), then \(\lim_{x\to - 1}f(x)=1\).

Step4: Find function value

The solid dot on the graph at \(x=-1\) is at \(y = - 2\). So, \(f(-1)=-2\).

Answer:

\(\lim_{x\to - 1^{-}}f(x)=1\)
\(\lim_{x\to - 1^{+}}f(x)=1\)
\(\lim_{x\to - 1}f(x)=1\)
\(f(-1)=-2\)