in this section, we will investigate indeterminate forms and an new technique for calculating limits...

In this section, we will investigate indeterminate forms and an new technique for calculating limits of such expressions.

Section 4.2 L’Hôpital’s Rule

Definition

A function having the property that as

it is true that ,

has an indeterminate form of type .

Definition

A function having the property that as

it is true that

has an indeterminate form of type .

For example:

Definition

A function having the property that as

it is true that ,

has an indeterminate form of type .

Definition

A function having the property that as

it is true that

has an indeterminate form of type .

For example:

Definition

A function having the property that as

it is true that ,

has an indeterminate form of type .

Definition

A function having the property that as

it is true that ,

has an indeterminate form of type .

For example:

TheoremL’Hôpital’s Rule

Let f and g be differentiable functions such that

has an indeterminate form of type or type .

Then:

Note the “a” above could be ±∞.

Example 1

Evaluate

Example 2

Evaluate

Example 3

Evaluate

Example 4

Evaluate

Example 5

Evaluate

Example 6

Evaluate

Example 7

Below is shown the graph of y = f(x). Find:

(a)

(b)

(c)