# De L’Hospital’s rule

Let at *x * *a * for

functions * f *( *x *) and * g* ( *x* ), *differentiable* in some neighborhood

of the point *a* , the conditions are executed:

This theorem is called ** de L’Hospital’s rule**. It allows to calculate limits of ratios of

functions, when both a numerator and a denominator approach either zero, or infinity. As mathematicians say,

*de*

L’Hospital’s rule

L’Hospital’s rule

*permits to get rid of indeterminacies of types*0 / 0 and

/ .

At indeterminacies of other types:

– ,

×0

, 0^{ 0} , ^{ 0},

it is necessary to do some *identical* transformations to reduce them to

one of these two indeterminacies:

either 0 / 0 , or / .

After this it is possible to use de L’Hospital’s rule. Show some of possible transformations of the above

mentioned indeterminacies.

1) | – :let f ( x ) , g ( x ) , then this indeterminacy is reduced to the type 0 / 0 by the following transformation: |

2) |
let f ( x ) , ( g x ) 0 , then this indeterminacy is reduced to the types 0 / 0 or /by the following transformations: |

3) | the rest of the indeterminacies are reduced to the first ones by the logarithmic transformation: |

If after using of de L’Hospital’s rule the indeterminacies of the types 0 / 0 or

/ remain, it is necessary to repeat it. The multifold use of de L’Hospital’s rule can

give the required result. The de L’Hospital’s rule is also applicable, if *x*

.