Finding Limits Analytically. Example Problem 1

Finding Limits Analytically. Example Problem 1

Find:
lim_(x->-4) (x^2 -6x-40)/(x^2 +6x+8)
lim_(x->-4) ((x+4)(x-10))/((x+4)(x-2))
lim_(x->-4) (x-10)(x-2)=-14/-2=7
We can find the limit of an algebraic expression in several ways. One way is to use simple
substitution, but that doesn't work for indeterminate forms; evaluating numerator and
denominator with negative 4 will yield 0/0.
To clear up the 0/0, we need to factor.
The top factors of 4x – 10 are 4x + 2 and 4x – 2 The denominator factors of 4x – 10 are 4x +
2 and 4x. Both the numerator and denominator have a factor of 4x + 2, so if you add these
two equations together, you will get a true statement.
Thus, as x approaches negative 4, we have the derivative of x minus 10 divided by x plus 2.
Because the numerator and denominator both equal we eliminate the problem of division
by 0.
We have taken the expression 0/0 out of the limit. Therefore, we are left with only the
numerator and denominator as fractions to simplify.

To evaluate the limit again, plug in
negative 4. We will evaluate it in the same way we evaluated it earlier by dividing the
numerator and denominator by 2. By doing so, we get -14/2, which is -7. The limit is -7.