Find the value of (x^2 - 4) / (x – 2) lim x-->2, using two different methods.

Here we require the answer to be derived using more than
one method.

First we can see that x^2 – 4 can be written as
(x - 2)*(x + 2)

So (x^2 - 4) / (x – 2) lim
x-->2

=> (x - 2)*(x + 2) / (x-2) lim
x-->2

=> (x + 2) lim
x-->2

=> 2 + 2 =
4

Next, we can use L’ Hopital’s Rule because
substituting x = 2 in the expression (x^2 - 4) / (x – 2) yields the form 0/0 which is
indeterminate. Therefore we can use [f’(x)/g’(x)] for x=2 instead of (x^2 - 4) / (x – 2)
lim x-->2.

Now f(x) = (x^2 - 4) => f’(x) =
2x

g(x) = x-2 => g’(x) =
1

Therefore [f’(x)/g’(x)] for
x=2

=> 2*2 /
1

=>
4

As we see,
both the methods give the same result.