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Evaluate the limit of the function (x^2-1)/(x-1) if x goes to 1?

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lesoir | Student, Undergraduate | (Level 3) eNoter

Posted July 28, 2011 at 1:03 AM via web

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Evaluate the limit of the function (x^2-1)/(x-1) if x goes to 1?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted July 28, 2011 at 1:08 AM (Answer #1)

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We'll substitute x by 1 in the expression of the function:

lim (x^2 - 1)/(x-1) = (1-1)/(1-1) = 0/0

x->1

Since we've get an indetermination, we can apply L'Hospital's rule:

lim f(x)/g(x) = lim f'(x)/g'(x)

Let f(x) = x^2 - 1 => f'(x) = 2x

Let g(x) = x - 1 => g'(x) = 1

lim (x^2 - 1)/(x-1) = lim 2x/1

x->1                       x->1

We'll replace x by 1:

lim 2x/1 = 2/1 = 2

The requested limit of the given function, if x approaches to 1, is lim (x^2 - 1)/(x-1) = 2.

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beckden | High School Teacher | (Level 1) Educator

Posted July 28, 2011 at 1:10 AM (Answer #2)

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Since `(x^2-1) = (x-1)(x+1)`

`lim_(x->1) (x^2-1)/(x-1) = lim_(x->1) ((x+1)(x-1))/(x-1) = lim_(x->1) (x+1) = 2`

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