Evaluate the limit of the function f(x) given by f(x)=(x^2-15x+14)/(x-1) if x goes to1?
We have to find the value of lim x-->1 [(x^2-15x+14)/(x-1)]
Substituting x =1, gives the indeterminate form 0/0.
So we can use l"Hopital's rule and interchange the numerator and denominator with their derivatives.
=> lim x-->1 [(2*x - 15)/1)
substitute x = 1
=> 2 - 15
The required limit is -13.
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