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L’Hôpital’s Rule (also written as L’Hospital Rule) is a means for evaluating limits that lead to an indeterminate form of either \(\textstyle\frac{\infty}{\infty}\) or \(\textstyle\frac{0}{0}\) by “simplifying” the function. L’Hôpital’s Rule can also be viewed as a measurement that analyzes the ratio of the rate of change between two given functions \(f(x),g(x)\). When applying this rule remember that you are NOT applying the quotient rule, rather you are inspecting the quotient of derivatives.
\(\underset{x\to a}{\lim}\frac{f(x)}{g(x)}\stackrel{\small{L’H}}{=}\underset{x\to a}{\lim}\frac{f'(x)}{g'(x)}\)
CAUTION: Although L’Hôpital’s Rule is useful a tool, it does not always lead us to the solution. to a solution. Some limits need to be evaluated algebraically. Consider the following examples.
(1) \(\displaystyle\underset{x\to\infty}{\lim}\frac{\texttt{e}^x+\texttt{e}^{-x}}{\texttt{e}^x-\texttt{e}^{-x}}\) (2) \(\displaystyle\underset{x\to\infty}{\lim}\frac{x}{\sqrt{x^2+1}}\) (3) \(\displaystyle\underset{x\to\infty}{\lim}\frac{x+\texttt{cos}(x)}{x-\texttt{cos}(x)}\) (4) \(\displaystyle\underset{x\to\infty}{\lim}\frac{x+\texttt{sin}(x)}{x}\) (An issue with oscillation)
Limits
Limits
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