Verify if the function has limit if x-->1 f(x)=(x^2-4x+3)/(x-1)

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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)

We'll simplify:

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

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