How to calculate limit of function with no use of derivatives? f(x)=(x^2+12x-13)/(x-1), x approaches 1

First, we'll substitute x by 1 and we'll verify if it is
an indetermination:

lim (x^2+12x-13)/(x-1) =
(1+12-13)/(1-1) = (13-13)/(0) = 0/0

Since we've get an
indetermination, that means that x = 1 represents a root for both numerator and
denominator.

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

1 + x =
-12

x = -12-1

x =
-13

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

x^2+12x-13 =
(x-1)(x+13)

We'll re-write the
limit

lim (x^2+12x-13)/(x-1) = lim
(x-1)(x+13)/(x-1)

We'll simplify inside
limit:

lim (x-1)(x+13)/(x-1)= lim
(x+13)

We'll substitute again x by
1:

lim (x+13) = 1 + 13 =
14

The limit of the function, if x approaches
to 1, is:lim (x^2+12x-13)/(x-1) = 14.