Question

lim ((x^2-64)/(x+8))
x->-8

Answer 1

In calculating the limit of a rational fraction, if the denominator does <i>not</i> evaluate to zero, the limit is found be simply substituting the limit of x for x.

If the denominator evaluates to zero, and so does the numerator, we can see if there is a common factor. If there is, we can cancel the common factor and proceed as another rational fraction (or a polynomial).

As x->8, the denominator evaluates to zero, so we check the numerator. The numerator also evaluates to zero.

We note that the numerator factors into (x+8)(x-8), which has a common factor of (x+8) with the denominator.

Cancelling the common factor leaves us with (x-8), which is a polynomial that can be evaluated by simple substitution:
Lim x->-8 (x-8) = -16.

Therefore
Lim x->-8 (x^2-64)/(x+8) = -16