Verify if the value of the limit of the function f(x)=(x^2-6x+5)/(x-1) is 4 x-->1

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

We'll substitute x by 1 in the expression of the
function.

lim f(x) = lim
(x^2-6x+5)/(x-1)

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

1 + x2
= 6

x2 = 6 - 1

x2 =
5

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

x^2-6x+5 =
(x-1)(x-5)

We'll re-write the
limit:

lim (x - 1)(x - 5)/(x -
1)

We'll simplify:

lim (x -
1)(x - 5)/(x - 1) = lim (x - 5)

We'll substitute x by
1:

lim (x - 5) = 1-5

lim
(x^2-6x+5)/(x-1) = -4

So, the limit of the
given function is -4 and not 4. lim (x^2-6x+5)/(x-1) = -4, for
x->1.