Evaluate the limit of the function f(x) given by f(x)=(x^2-15x+14)/(x-1) if x goes to1?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find the value of lim x-->1 [(x^2-15x+14)/(x-1)]

Substituting x =1, gives the indeterminate form 0/0.

So we can use l"Hopital's rule and interchange the numerator and denominator with their derivatives.

=> lim x-->1 [(2*x - 15)/1)

substitute x = 1

=> 2 - 15

=> -13

The required limit is -13.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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

lim f(x) = lim (x^2-15x+14)/(x-1)

lim (x^2-15x+14)/(x-1) =  (1-15+14)/(1-1) = 0/0

We'll 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 = -(-15)/1

1 + x2 = 15

x2 = 15 - 1

x2 = 14

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

x^2-15x+14 = (x-1)(x-14)

We'll re-write the limit:

lim (x-1)(x-14)/(x - 1)

We'll divide by (x-1):

lim (x-1)(x-14)/(x - 1) = lim (x - 14)

We'll substitute x by 1:

lim (x - 14) = 1-14

lim (x - 14) = -13

The limit of the given function f(x) = (x^2-15x+14)/(x-1) is:

lim (x^2-15x+14)/(x-1) = -13

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