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

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william1941 | College Teacher | (Level 3) Valedictorian

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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.

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neela | High School Teacher | (Level 3) Valedictorian

Posted on

To find lt x-->2 (x^2-4)/(x-2).

We put x= 2 in (x^2-4)/(x-2) and we get (4-4)/(2-2) = 0/0 form of indeterminate value.

Therefore x=2 makes both numerator and denominator zero. So x-2 is a factor of both numerator and denominator. We divide both numerator and denominator by x-2. And then take the limit as x--> 2 on both sides:

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

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

Now we take the limit as x--> 2.

Lt x-->2 (x^2-4)/(x-2) = Lt x--> 2 ,   (x+2) =  2+2 = 4.

So Lt x--> 2 (x^2- 4)/(x-2) = 4.

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