L'Hopital's Rule

Read this section to learn how to use and apply L'Hopital's Rule. Work through practice problems 1-3.


When we began taking limits of slopes of secant lines, \mathrm{m}_{\mathrm{sec}}=\frac{\mathrm{f}(\mathrm{x}+\mathrm{h})-\mathrm{f}(\mathrm{x})}{\mathrm{h}} as \mathrm{h} \rightarrow 0, we frequently encountered one difficulty: both the numerator and the denominator approached 0. And since the denominator approached 0, we could not apply the Main Limit Theorem. In each case, however, we managed to get past this " 0/0 " difficulty by using algebra or geometry or trigonometry, but there was no common approach or pattern. The algebraic steps we used to evaluate \lim \limits_{h \rightarrow 0} \frac{(2+h)^{2}-4}{h} seem quite different from the trigonometric steps needed for \lim \limits_{h \rightarrow 0} \frac{\sin (2+h)-\sin (2)}{h}.

In this section we consider a single technique, called l'Hô pital's Rule (pronounced Low-Pee-Tall), which enables us to quickly and easily evaluate limits of the form " 0/0 " as well as several other difficult forms.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2011/11/4-7LHopitalsRule.pdf
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