Derivatives in Function Analysis - De L'Hopital's Theorem

When we have a limit that leads to an indeterminate form that we can't get around, such as:
We can instead take the limit of the ratio of the derivatives of these functions, instead of the function themselves. This is De L'Hopital's theorem.

You may attempt to determine the limit of an indeterminate form by taking the limit of the ratio of the derivatives of the functions involved in this indeterminate form

Example:

Consider the following function, and compute the limit for .
Wanna try substituting?
Can't really manipulate it either.
So differentiate each component of the fraction. Careful not to treat it as a difference rule, but as just a ratio of derivatives!

In this case we have easily proven the fundamental limit:

Easy peasy. Remember this. Seriously. It's gonna save you.

Now,
Grand Review - Advanced Differentiation, Function Analysis and De L'Hopital's Theorem.