Question

Find the limit of the function algebraically.

Answer 1

\[\lim_{x \rightarrow -2} \frac{ x ^{2} - 4}{ x + 2 }\]

Answer 2

the problem here is that if you plug in -2 you get the undefined form of 0/0. So you need to transform the expression you want to take the limit from. There are several ways to do that. One of the first tricks is to fraction out the x's. That is a neat trick when you have the limit to infinite. But in this case that won't help. As a last resort there is L'Hospital's rule.

L'Hospital's rule says that if we take the limit of f(x)/g(x) that is equal to taking the limit of f'(x)/g'(x). So you need to take the derivatives of the numenator and denominator.

\[\lim_{x \rightarrow -2} \frac{ 2x }{ 1}\]

Can you now solve the limit?

Answer 3

-2x * 2(x*1/1) = -4 ?

Answer 4

It's just 2*(-2)/1=-4. You plug the value of -2 where the x is.

Answer 5

Oh okay. And is that all or are there more steps? I think -4 is the final answer, correct?

Answer 6

Yes.

Basically limits are easy: just plug the value in the variable x and calculate. But the problem is that there are some limits you cannot solve, these are 0/0, infinity/infinity, 0/infinity, infinity - infinity and 1^infinity. When you end up with an undefined result after just plugging in the value for x, you need to transform the expression algebraically so that you can plug in the values with the next step. One of those tricks is L'Hospital's rule, but it is the last resort trick.