Applications of differentiation, L' Hospital's rule - limits of indeterminate forms

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Differential calculus - derivatives

Applications of differentiation

L' Hospital's rule - limits of indeterminate forms

The rule, based on the generalized mean value theorem, that enables the evaluation of the limit of an indeterminate quotient of functions as the quotient of limits of their derivatives is called L'Hospital's rule.

a) Indeterminate forms

Let f and g are differentiable functions near a (except possibly at a) and g'(x) is not 0, and if

or

then

This rule can also be applied in case when a = oo.

If the quotient of the derivatives repeats an indeterminate form then we can proceed with the quotients of successive derivatives, provided required conditions hold.

b) Indeterminate form 0 · oo

Write the product f (x) · g (x), where

as a quotient

c) Indeterminate form oo - oo

Rewrite difference f (x) - g (x) into the product

and first solve the indeterminate

form

If

then, write the above product as

d) Indeterminate (or indefinite) powers, 1^oo, 0⁰ and oo⁰

we solve taking the natural logarithm of the given expression by writing

f (x)^{g
(x)} = y or ln y = g (x) · ln f (x)

so it then becomes an indeterminate product.

There are cases where we should combine both L'Hospital's rule and elementary methods to evaluate limits.

Applications of L'Hospital's rule - evaluation of limits of indeterminate forms, examples

Example: Evaluate the following limit

Solution:

Example: Evaluate the following limit

Solution:

Example: Evaluate the following limit

Solution:

Example: Evaluate the following limit

Solution:

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