Removable discontinuity?

Question

A function f(x) is said to have a removable discontinuity at x=a if:
1. f is either not defined or not continuous at x=a.
2. f(a) could either be defined or redefined so that the new function IS continuous at x=a.

 

Let f(x) = (2x^2+3x-44)/(x-4)

 

Show that f(x) has a removable discontinuity at x=4 and determine what value for f(4) would make f(x) continuous at x=4.

 

must define f(4)= ???

Answer 1

f(x) = (2x² + 3x - 44)/(x - 4) = [(x - 4)(2x + 11)]/(x -4)

                                                    = 2x + 11, if x ≠ 4

 

When x = 4, 2x + 11 = 19

 

The graph of y = f(x) is the same as the graph of

y = 2x + 11, except for a "hole" (removable discontinuity) at the point (4, 19).  

 

To make f(x) continuous at x = 4, we must fill in the hole.

So, we define f(4) to equal 19.