3. the graph of a function f(x) is pictured below. determine all x - values at which f(x) has a…

3. 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

3. 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

Answer

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)