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* *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.

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*
.