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« Limits of Trigonometric Functions
Limits: (lesson 5 of 5)

Lhospital Rule

L'Hospital's Rule for L'Hospital Rule


Suppose . Then

If L' Hospital Rule, then solution.

If exampletends to math solutionin the limit, then so does example.

Here is a case of solution.


Examples 1:

L'Hospital Rule example

Examples 2:

L'Hospital Rule solution

Examples 3:

example3

Example4:

Sometimes it is necessary to use L'Hospital's Rule several times in the same problem.

example 4

Example5:

Here is a more elaborate example involving the indeterminate form 0/0. Applying the rule a single time still results in an indeterminate form. In this case, the limit may be evaluated by applying L'Hôpital's rule three times:

example 5

solution

solution

L'Hospital Rule

L'Hospital Rule

formula

L'Hospital's Rule for limit


Suppose . Then

If L'Hospital Rule, then calculus.

If limit tends to limit example in the limit, then so does solution.

Here is a case of example

Example 6:

example 6

Example 7:

example 7

Iterate the above until the exponent is 0. Then one sees that the limit is 0.

Example 8:

This one also involves L'Hospital Rule example 8:

L'Hospital Rule solution 8

Basic indeterminate forms (all others reduce to these):

indeterminate forms

Other indeterminate forms:

indeterminate forms

Example 9:

To handle a case of indeterminate forms, the difference of two functions is converted to a quotient:

L'Hospital Rule

L'Hospital Rule

solution

solution

Example 9:

To handle a case of example 9, the difference of two functions is converted to a quotient:

calculus

When NOT to use L'Hospital's rule

L'Hospital Rule

because L'Hospital Rule