Whhich is the limit of the function lim (x^3-1)/(x-1), x-->1? Explain the technique used.

In order to evaluate the limit, we'll choose the dividing
out technique.

We'll apply the direct substitution, by
substituting the unknown x, by the value1 and we'll see that it fails, because both,
numerator and denominator, are cancelling for x=1.

Also,
because x=1 is a root for both, that means that (x-1) is a common facor for
both.

We'll write the numerator using the
formula:

a^3-b^3=(a-b)(a^2+ab+b^2)

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

Now,
we'll evaluate the limit:

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

Now, we can divide out like
factor:

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

We can apply the replacement theorem and we'll
get:

lim (x^2+x+1) = 1^2 + 1 + 1 =
3

So, lim (x^3-1)/(x-1) =
3.