Limit of (x)/(x^2-4) as x approaches 2

There is no reason for extra brackets around the x. Furthermore. none of the answerers have given you a correct, definitive answer:

Lim x → 2 x / (x² - 4)
Lim x → 2 2 / 0

Lim x → 2⁺ x / (x² - 4) = +∞
Lim x → 2⁻ x / (x² - 4) = -∞

Since the limit from the right and left are not equal the limit as x → 2 x / (x² - 4) does not exist.

DNE

limit x/(x^2 - 4) = 2/(4-4) = 2/0 = ±∞
x->2

depending on whether you are approaching 2 from the left (-∞) or from the right (+∞)

You cannot use l'Hopital's rule here, since we do not have indeterminate
form 0/0 or ±∞/∞

When NOT to use l'Hôpital's rule

The most important thing to learn about l'Hôpital's rule is when it should not be used:
When the limits of the two parts are not both 0, or both infinite. In this case the rule is likely to give a wrong answer!

but yeah..
x/(x+2)(x-2)
..

apply L hospital's rule
then it turns out as 1/2x
as x approaches 2 it'll be 1/4

x/(x^2-4)=x/(x^2-2^2)=x/((x-2)(x+2)). Therefore lim x/((x-2)(x+2)) as x->2=2/(2-2)(2+2)=2/0=infinity.