Tuesday, June 12, 2012

Evaluate the limit of the function f(x) given by f(x)=(x^2-15x+14)/(x-1) if x goes to1?

First, we'll verify if the limit exists, for x = 1,
so, we'll replace x by 1.


lim f(x) = lim
(x^2-15x+14)/(x-1)


lim (x^2-15x+14)/(x-1) = 
(1-15+14)/(1-1) = 0/0


We'll calculate the roots of the
numerator. Since x = 1 has cancelled the numerator, then x = 1 is one of it's 2
roots.


We'll use Viete's relations to determine the other
root.


x1 + x2 = -(-15)/1


1 +
x2 = 15


x2 = 15 - 1


x2 =
14


We'll re-write the numerator as a product of linear
factors:


x^2-15x+14 =
(x-1)(x-14)


We'll re-write the
limit:


lim (x-1)(x-14)/(x -
1)


We'll divide by (x-1):


lim
(x-1)(x-14)/(x - 1) = lim (x - 14)


We'll substitute x by
1:


lim (x - 14) = 1-14


lim (x
- 14) = -13


The limit of the given function
f(x) = (x^2-15x+14)/(x-1)
is:


lim (x^2-15x+14)/(x-1) =
-13

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