Question

1. the graph of a function f(x) is pictured below. determine all x - values at which f(x) has a discontinuity and classify each as either removable, jump, or in justify your answers. see example 1.8.6 in chads calculus notes

Explanation:

Step1: Recall discontinuity types

Removable discontinuity has a hole, jump discontinuity has a jump in the graph, and infinite discontinuity has a vertical - asymptote.

Step2: Analyze \(x = - 1\)

As \(x\to - 1\), the left - hand limit and the right - hand limit exist but are not equal. The function has a jump at \(x=-1\). So it is a jump discontinuity.

Step3: Analyze \(x = 1\)

There is a hole in the graph at \(x = 1\). The limit as \(x\to1\) exists, but the function is not defined at \(x = 1\). So it is a removable discontinuity.

Step4: Analyze \(x = 3\)

As \(x\to3\), the function approaches \(\pm\infty\). There is a vertical asymptote at \(x = 3\). So it is an infinite discontinuity.

Answer:

Discontinuities at \(x=-1\) (jump), \(x = 1\) (removable), \(x = 3\) (infinite)