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

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to calculate the value of lim x-->1 [(x^2+12x-13)/(x-1)].

We cannot substitute x = 1 directly as that yields an indeterminate form. Instead we do the following.

lim x-->1 [(x^2+12x-13)/(x-1)]

=> lim x-->1 [(x^2 +13x - x -13)/(x-1)]

=> lim x-->1 [(x(x +13) - 1(x + 13))/(x-1)]

=> lim x-->1 [(x - 1)(x +13)/(x-1)]

=> lim x-->1 [(x +13)]

substitute x = 1

=> 1 + 13

=> 14

The required value of lim x-->1 [(x^2+12x-13)/(x-1)] = 14

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

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