In the last post we talked about why L'Hospital's rule might help you to help you evaluate indeterminate forms that you encounter on the AP Calculus exam. Here we will look at some L Hospital's Rule Examples and how the rule can be used to check whether these indeterminate forms actually evaluate to some finite quantity.
A tip – it is most unlikely that your final answer to a problem on the AP Calculus exam should be an indeterminate form. Note that 0 or infinity themselves are not indeterminate forms, and are perfectly acceptable as answers. Indeterminate forms are things like 0/0 or ∞/∞ or 0*∞ or 00 or ∞0.
Let us look at an example of how L'Hospital's rule can be used to evaluate an indeterminate form. Before we go further, recall that L'Hospital's rule applies to limits. If your indeterminate form is not already a limit, you can usually find a way to write it as a limit before applying L'Hospital's rule.
Consider the limit : both numerator and denominator tend to infinity as x tends to infinity. However, we know that ex grows faster than x2, so it seems quite likely that our final answer should be infinity. That is our intuition, but can we prove it?
Hmmm....still an indeterminate form. Well, we know we can use L'Hospital's rule on indeterminate forms, so let us apply it to this new indeterminate expression. Once again the derivative of the numerator is just ex, while that of the denominator is 2.
If we now take the limit as x tends to infinity, we end up with ∞/2 = ∞. Thus our intuition was proved correct.
Note that even though the first application of L'Hospital's rule resulted in another indeterminate form, we simply applied the rule to the new indeterminate form and this time met with success.
Sometimes more than two iterations are required, but it should soon be apparent whether you are going in the right direction. Winners never quit, and quitters never win, but if you never win and never quit, perhaps you should reevaluate your strategy....