Wednesday, May 23, 2012

What is the limit of the function (x^4-16)/(x-2), if x goes to 2?

We notice that the numerator is a difference of squares that
returns the product:

x^4-16 =
(x^2-4)(x^2+4)


We notice that the 1st factor is also
a difference of squares that returns the product:


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



We'll re-write the
function:


(x^4-16)/(x-2) =
(x-2)(x+2)(x^2+4)/(x-2)



We'll
simplify:


(x^4-16)/(x-2) =
(x+2)(x^2+4)



We'll evaluate the limit the
fraction:


lim (x^4-16)/(x-2) = lim
(x+2)(x^2+4)



We'll replace x by
2:


lim (x+2)(x^2+4) = (2+2)(4+4) = 4*8 =
32



The value of the limit, if x
approaches to 2, is lim (x^4-16)/(x-2) = 32.

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