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What is limit of `(x^3 - 1)/(x-1)`  if x goes to 1?

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greenbel | (Level 2) Honors

Posted July 16, 2013 at 4:49 PM via web

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What is limit of `(x^3 - 1)/(x-1)`  if x goes to 1?

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ishpiro | Teacher | (Level 2) Associate Educator

Posted July 16, 2013 at 5:19 PM (Answer #1)

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To calculate `lim_(x->1) (x^3-1)/(x-1)` , factor the numerator of the fraction as a difference of two cubes:

`(x^3 - 1)/(x - 1) = ((x-1)(x^2 + x + 1))/(x-1) = x^2 + x + 1`

The resultant function is continuous at x = 1, so the limit can be evaluated by plugging in x = 1:

`lim_(x->1) (x^2 + x + 1) = 1^2 + 1 + 1 = 3`

This limit equals 3.

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