Calculate lim (x-2)/(x^2-4), x->2.

To calculate the limit, we'll apply the dividing out
technique.

We'll apply the direct substitution, by
substituting the unknown x, by the value 2 and we'll see that it fails, because both,
numerator and denominator, are cancelling for x=2. That means x=2 is a root for both,
that means that (x-2) is a common factor for both.

We'll
write the denominator using the
formula:

a^2-b^2=(a-b)(a+b)

x^2-4
= (x-2)(x+2)

Now, we'll evaluate the
limit:

lim (x-2)/(x^2-4)= lim
(x-2)/(x-2)(x+2)

Now, we can divide out like
factor:

lim (x-2)/(x^2-4) = lim
1/(x+2)

We can apply the replacement theorem and we'll
get:

lim 1/(x+2) = 1/(2+2) =
1/4

So, lim (x-2)/(x^2-4) =
1/4