Plz help with this function question?

Question

determine whether the function x-2/x^2-4 is continuous at x=2

Answer 1

ANSWER: Continuous and the value for x= 2 is 0.25

2 WAYS TO EXPLAIN THAT:

1)

If it's continuous, then the limit when x-->2 both from the left side (x--->2-) and from the right side (x-->2+) must be equal:

a) For lim x--->2- let's give to x the value 1.99999

the value of the function for x=1.99999 is (1.99999-2)/(1.99999^2-4) = 0.2500001

b) For lim x--->2+ let's give to x the value 2.00001

the value of the function for x=2.00001 is (equal to 0.249999)

So the value of the function for x=2 is 0.25 and the function is continuous.

2) You could also have used the theorem of L'Hopital differentiating the equation both the upper and the lower part:

lim x-->2 [x-2/x^2-4] = lim x--->2 [1/2x] = 1/4 = 0.25

Answer 2

The wording is wrong. Given fraction itself in undefined at 2. How can you comment on continuity?

If f(x) = x-2/x^2-4 , if x=/= 2

        = k if x = 2

this can be said to be cotinuos if k = 1/4 and discontinuous otherwise.

Some have attempted as it is!!! Hat's off to them!