is it possible for equation to have a removable and infinite discontinuity

Question

Given f(x)= 4x^3-36x-4x^2+36/2-3x+x^2

Is it possible for f(x) to have a removable AND and infinite discontinuity at the same point (value of a). Offer support to your answer as to why you think your decision/answer is correct. Give mathematic examples

Answer 1

A function f has a removable discontinuity at x = x0 (a point in its domain, i.e. f(x0) exists) if two things are true:

1. The limit as x -> x0 of f(x) = L exists and is finite

2. f(x0) isn't equal to L

 

A function f has an infinite discontinuity at x = x0 (a point in its domain, i.e. f(x0) exists) if either (or both):

1. The limit as x -> x0 from the left of f(x) doesn't exist

2. The limit as x -> x0 from the right of f(x) doesn't exist

 

Do you think it's possible for a function to exhibit both of these properties at the same point, or are they mutually exclusive?