To 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^2-4x+3)/(x-1)
lim (x^2-4x+3)/(x-1) = (1-4+3)/(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 = -(-4)/1
1 + x2 = 4
x2 = 4 - 1
x2 = 3
We'll re-write the numerator as a product of linear factors:
x^2-4x+3 = (x-1)(x-3)
We'll re-write the limit:
lim (x-1)(x-3)/(x - 1)
lim (x-1)(x-3)/(x - 1) = lim (x - 3)
We'll substitute x by 1:
lim (x - 3) = 1-3
lim (x^2-4x+3)/(x-1) = -2