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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