a calculus limit problem?

Question

lim (x^3-1)/(x^2-1)=?
x->1 from the right
x->1 from the left
x-> 1

Answer 1

lim (x^3 - 1) / (x^2 - 1)
x -> 1+

Factor the numerator and denominator.

lim (x - 1)(x^2 + x + 1) / (x - 1)(x + 1)
x -> 1+

Cancel (x - 1) from numerator and denominator.

lim (x^2 + x + 1) / (x + 1)
x -> 1+

Now, plug in the value x = 1.

(1^2 + 1 + 1) / (1 + 1) = 3/2

We're going to get the same value of the limit from the left, and for the limit itself.

Answer 2

lim x→1 of (x³ - 1)/(x² - 1)
= lim x→1 of (x - 1)(x² + x + 1)/{(x - 1)(x + 1)}
= lim x→1 of (x² + x + 1)/(x + 1)
= (1 + 1 + 1)/(1 + 1) = 3/2

It doesn't matter which side you approach it from, the limit is the same.

Answer 3

Limits are the first step to learning differentation... iirc...

There are three number you need to plug in to address this problem: 0,1 and 2.

at 0 you get -1/-1 = 1
at 1 you get 0/0 = infinity
at 2 you get 7/3= 2.33
at .1 you get -.999/-.99 = 1.09

As you see, both go to positive infinity. You can prove this to yourself with a spreadsheet.