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