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De L'Hospitalís rule

Let at x 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'Hospitalís 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'Hospitalís rule permits to get rid of indeterminacies of types 0 / 0 and / .

At indeterminacies of other types: Ė , ×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'Hospitalís rule. Show some of possible transformations of the above mentioned indeterminacies.

1)   Ė :
 let   f ( x ) , g ( x )   , then this indeterminacy is reduced to the type 0 / 0 by the following  transformation:
2)

  × 0 :
 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'Hospitalís rule the indeterminacies of the types 0 / 0 or / remain, it is necessary to repeat it. The multifold use of de L'Hospitalís rule can give the required result. The de L'Hospitalís rule is also applicable, if  x .

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