how would you "remove discontinuity" of f? in other words, how would you define f(4) in order to make f continuous at 4

Question

How would you "remove discontinuity" of f? in other words, how would you define f(4) in order to make f continuous at 4

given f(x) = (x^2-3x-4)/(x-4)

f(4) = ?

Please provide detailed explanation/work

Answer 1

Hello James,

The function f(x) = (x²-3x-4)/(x-4) is discontinuous at x = 4 because if we plug 4 into this equation, we get 0/0, which is undefined because we can't have 0 in the denominator. However, 0/0 also means the limit lim_x→4 f(x) exists. To find this limit, we simplify (x²-3x-4)/(x-4) by factoring:

x² - 3x -4 = (x+4)(x-1); so (x²-3x-4)/(x-4) = x - 1 and lim_x→4 f(x) = lim_x→4 (x-1) = 4 - 1 = 3

Therefore, to remove the discontinuity, we define f(4) = 3.

I am happy to discuss this further if you have more questions.

Best,

Kevin

Answer 2

If you factor the numerator, you get:

f(x) = (x-4)(x+1)/(x-4)

The (x-4)'s cancel out, leaving

f(x) = x+1

f(4) = 5