Question

in the following exercises, use the graph of the function ( y = f(x) ) shown here to find the values, estimate when necessary.

1. ( lim_{x \to 0^{-}} f(x) )
2. ( lim_{x \to 0^{+}} f(x) )
3. ( lim_{x \to 0} f(x) )

Explanation:

Step1: Analyze the left - hand limit ($x\to0^{-}$)

When \(x\) approaches \(0\) from the left - hand side (\(x\to0^{-}\)), we look at the part of the graph for \(x < 0\). As \(x\) gets closer and closer to \(0\) from the left (\(x<0\)), the \(y\) - value of the function approaches \(1\). So, \(\lim_{x\to0^{-}}f(x)=1\)

Step2: Analyze the right - hand limit ($x\to0^{+}$)

When \(x\) approaches \(0\) from the right - hand side (\(x\to0^{+}\)), we look at the part of the graph for \(x>0\). As \(x\) gets closer and closer to \(0\) from the right (\(x > 0\)), the \(y\) - value of the function approaches \(-4\). So, \(\lim_{x\to0^{+}}f(x)=-4\)

Step3: Analyze the two - sided limit ($x\to0$)

The two - sided limit \(\lim_{x\to0}f(x)\) exists if and only if \(\lim_{x\to0^{-}}f(x)=\lim_{x\to0^{+}}f(x)\). Since \(\lim_{x\to0^{-}}f(x)=1\) and \(\lim_{x\to0^{+}}f(x)=-4\), and \(1
eq - 4\), we have \(\lim_{x\to0}f(x)\) does not exist.

Answer:

1. \(\lim_{x\to0^{-}}f(x)=1\)
2. \(\lim_{x\to0^{+}}f(x)=-4\)
3. \(\lim_{x\to0}f(x)\) does not exist.