Solve the limit of the function and do not use derivatives. f(x)=(x^2-6x+5)/(x^2-25), x approaches to 5

First, we'll substitute x by 5 and we'll verify if it is
an indetermination:

lim (x^2-6x+5)/(x^2-25) =
(5^2-6*5+5)/(5^2-25) = (30-30)/(25-25) = 0/0

Since we've
get an indetermination, that means that x = 5 represents a root for both numerator and
denominator.

We'll determine the 2nd root of the numerator,
using Viete's relations:

5 + x =
6

x = 6 - 5

x =
1

We'll rewrite the numerator as a product of linear
factors:

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

We notice that the denominator is a difference
of 2 squares and we'll write it as a product.

x^2 - 25 =
(x-5)(x+5)

We'll re-write the
limit

lim (x^2-6x+5)/(x^2-25) = lim
(x-1)(x-5)/(x-5)(x+5)

We'l simplify inside
limit:

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

We'll substitute again x by
5:

lim (x-1)/(x+5) = (5 - 1)/(5+5) = 4/10 =
2/5

The limit of the function, if x
approaches to 5, is: lim (x^2-6x+5)/(x^2-25) =
2/5.