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Homel'H\^opital's rule

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# l’Hôpital’s rule

L’Hôpital’s rule states that given an unresolvable limit of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, the ratio of functions $\frac{f(x)}{g(x)}$ will have the same limit at $c$ as the ratio $\frac{f^{{\prime}}(x)}{g^{{\prime}}(x)}$. In short, if the limit of a ratio of functions approaches an indeterminate form, then

$\lim_{{x\rightarrow c}}\frac{f(x)}{g(x)}=\lim_{{x\rightarrow c}}\frac{f^{{% \prime}}(x)}{g^{{\prime}}(x)}$ |

provided this last limit exists. L’Hôpital’s rule may be applied indefinitely as long as the conditions are satisfied. However it is important to note, that the nonexistance of $\lim\frac{f^{{\prime}}(x)}{g^{{\prime}}(x)}$ does not prove the nonexistance of $\lim\frac{f(x)}{g(x)}$.

Example: We try to determine the value of

$\lim_{{x\to\infty}}\frac{x^{2}}{e^{x}}.$ |

As $x$ approaches $\infty$ the expression becomes an indeterminate form $\frac{\infty}{\infty}$. By applying L’Hôpital’s rule twice we get

$\lim_{{x\to\infty}}\frac{x^{2}}{e^{x}}=\lim_{{x\to\infty}}\frac{2x}{e^{x}}=% \lim_{{x\to\infty}}\frac{2}{e^{x}}=0.$ |

Another example of the usage of L’Hôpital’s rule can be found here.

## Mathematics Subject Classification

26A24*no label found*26C15

*no label found*

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## Corrections

explanation by matte ✘

Accent by bbukh ✓

refer to example by archibal ✓

Accent by bbukh ✓