Checking If limits exist

[mikexz]

I know that the basic method of solving for a limit is to simplify (conjugate, etc..) a function so that you can plug in the value that x approaches. My problem is when I am working with very long/large functions, I often make small mistakes when simplifying (e.g. forget a negative) so I was wondering if there was a quick method for me to check (so I can be 100% sure) whether a function does or does NOT have a limit at a specific x value.

thanks!

 

[jwpaine]

Have you learned l'Hopital's rule?

Suppose that \(\displaystyle \lim_{x->a}f(x) = 0\) and \(\displaystyle \lim_{x->a}g(x) = 0\)

and suppose also that \(\displaystyle \lim_{x->a} \frac{f'(x)}{g'(x)}\) exists. Then \(\displaystyle \lim_{x->a} \frac{f(x)}{g(x)}\) exists, and

\(\displaystyle \lim_{x->a} \frac{f(x)}{g(x)} = \lim_{x->a} \frac{f'(x)}{g'(x)}\)

If you have not learned differentiation yet, than all you can do is try and factor, multiply by a conjugate, manipulate, and then make a peanut butter & jelly sandwich

Or... to prove that a limit exists at a certain value, you can do a proof using the formal epsilon-delta definition of the limit.... but it may be challenging to learn the method on your own if you have not been taught it.

Cheers,
John