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