Show that f(x) has a removable discontinuity

Question

Let f(x)=(2x²+4x-16)/(x-2)

show that f(x) has a removable discontinuity at x=2 and determine what value for f(2) would make f(x) continuous at x=2

must define f(x)= 

Answer 1

The numerator of the rational function f(x) can be factored as (2x +8) (x-2).   

The factor (x-2) is cancelled by the (x-2) in the denominator   to give

 

 2x + 8     which is continuous  and equal to 12  when x =2.

 

The original function is undefined at  x =2 because of the zero divided by zero condition at x =2.

This problem can be removed by simply adding the condition that  f(2) = 12.   to the function definition.

 

This function is an example of a function with a hole   (at x = 2 in this case).