Evaluate the limit of the function f(x)=(x^2-7x+6)/(x-1) , x-->1 .

First, we'll verify if the limit exists, for x = 1,
so, we'll substitute x by 1 in the expression of the
function.

lim f(x) = lim
(x^2-7x+6)/(x-1)

lim (x^2-7x+6)/(x-1) =  (1-7+6)/(1-1) =
0/0

We notice that we've get an indetermination
case.

We could apply 2 methods for solving the
problem.

The first method is to 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 = -(-7)/1

1 + x2
= 7

x2 = 7 - 1

x2 =
6

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

x^2-7x+6 =
(x-1)(x-6)

We'll re-write the
limit:

lim (x-1)(x-6)/(x -
1)

We'll divide by (x-1):

lim
(x-1)(x-6)/(x - 1) = lim (x - 6)

We'll substitute x by
1:

lim (x - 6) = 1-6

lim (x -
6) = -5

The limit of the given function f(x) =
(x^2-7x+6)/(x-1) is:

lim (x^2-7x+6)/(x-1) =
-5