Method: L’Hospital’s Rule

Step 1: Confirm that your limit problem results in an indeterminate form, and state that L’Hospital’s Rule applies.

• Most often this means applying limit Option 1, Plug It In.
• The $\frac{\infty }{\infty }$ result is usually something you need to be able to conceptually understand and state. They are not looking for any real work to prove it as it is usually the result of a limit heading to infinity , $\underset{x\to \infty }{\mathit{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\infty }{\infty }$. Think about the two equations, top and bottom, separately and consider what is happening as each gets larger, heads to infinity .

Step 2: Take the derivative of the top equation, $f\left(x\right)$, and the bottom equation,$\mathit{ g}\left(x\right)$ , individually to find your $f‘\left(x\right)$ and $g‘\left(x\right)$.

Step 3: Plug your derivatives into your original limit statement, and try to evaluate the limit again using your standard limit options.

• If you get an answer, great, you have your answer and you are done with the problem.
• If you get an indeterminate form again, then repeat this process as many times as you need until you do not get an indeterminate form.