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Example application of l'Hôpital's rule to f(x)=sin(x) and g(x)=−0.5x: the function h(x) = f(x)/g(x) is undefined at x = 0, but can be completed to a continuous function on whole by defining h(0) = f′(0)/g′(0) = −2.

L'Hôpital's rule is a rule in calculus that helps find limits using derivatives. The name of the rule comes from the mathematician Guillaume de l'Hôpital[1]. L'Hôpital's rule states that for any function f whose limit as f approaches some point c is an indeterminate form the value of the limit of the function is equal to the derivative of the bottom and the top.[2][3] It is shown as follows:

Examples

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Given the limit first plug in 5 to try to find the limit. Plugging in 5 will give an indeterminate form. Then L'Hôpital's rule is applied to give . Then simply plugging in 5 again yields the answer of 10.

The next example involves the sine function

plugging in 0 gives an indeterminate form

L'Hôpital's rule is applied and the derivatives of the top and bottom are taken

Plugging in 0 to the new expression gives 1, the answer to the limit

This example involves the natural log function

Plugging in 1 to the expression gives an indeterminate form

L'Hôpital's rule is applied and the derivatives of the top and bottom are taken.

expression is simplified

Plugging back in 1 gives the answer to the limit as 1/2

The next example includes both sine and natural log functions

Plugging in 1 to the expression gives an indeterminate form

L'Hôpital's rule is applied and the derivatives of the top and bottom are taken.

expression is simplified

Plugging back in 1 gives the answer to the limit as 2

The final example involves the tangent function

Plugging in 0 to the expression gives an indeterminate form

L'Hôpital's rule is applied and the derivatives of the top and bottom are taken.

Plugging back in 0 gives the answer to the limit as 1

References

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  1. ^ "De_LHopital biography". www-history.mcs.st-and.ac.uk. Retrieved 23 February 2017.
  2. ^ "Calculus I - L'Hospital's Rule and Indeterminate Forms". tutorial.math.lamar.edu. Retrieved 23 February 2017.
  3. ^ Guillaume, l'Hôpital (1696). Analyse Des Infiniment Petits, Pour L'intelligence Des Lignes Courbes. France.

See also

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Category:Articles containing proofs Category:Theorems in calculus Category:Theorems in real analysis Category:Limits (mathematics)