MAT 435

CHAPTER 2 : INDETERMINATE FORM AND IMPROPER INTEGRAL

2.1: L’HOPITAL RULE

Suppose that f and g are differentiable functions on an open interval containing

x = a, except possibly at x = a, and that and .

If has a finite limit, or if this limit is , then

Moreover this statement is also true in the case of a limit as .

In L’Hopital Rule, the numerator and denominator are differentiated separately, which is not the

same as differentiating .

The L’Hopital three-step process:

Step 1: Check that the limit of is an indeterminate form. If it is not, then

L’Hopital Rule cannot be used.Step 2: Differentiate f and g separately

Step 3: Find the limit of . If this limit is finite, , then it is equal to the

limit of .

2.2: INDETERMINATE FORM

A. INDETERMINATE FORM OF TYPE .

In these limits and , the numerator and denominator both approach zero.

This limit is called indeterminate forms of type . The technique for this limit is by using

L’Hopital Rule.

Example 2.2.1Use L’Hopital Rule to evaluate:

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a)

b)

Example 2.2.2

Evaluate a)

b)

c) 20x x

xcos1lim

d)

e)

f)

B. INDETERMINATE FORM OF TYPE .

An indeterminate form of type is a limit in which

.

Some examples are: ,

,

Example 2.2.3

Evaluate a)

b)

C. INDETERMINATE FORM OF TYPE

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A limit of a product , is called an indeterminate form of type if

.

Frequently limit problems of this type can be converted to the form of by writing:

.

Then the limit can be treated using L’Hopital Rule.

Example 2.2.4

Evaluate a)

b)

D. INDETERMINATE FORM OF TYPE .

Limits of the form give rise to indeterminate forms of type .

All three types are treated by introducing a dependent variable:

And then calculating:

=

Once the value of is known, the value of is also

known.

Example 1.3.1

Show that a)

b)

E. INDETERMINATE FORM OF TYPE .This form can be treated by combining the two terms into one and then use the L’Hopital Rule to solve.

Example 2.2.6

Evaluate a)

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b)

2.3 IMPROPER INTEGRALS

The objective is to extend the concept of a definite integral to allow for infinite intervals of integration and integrands with vertical asymptotes within the interval of integration.

The vertical asymptote is call infinite discontinuities. Integral with infinite intervals of integration or infinite discontinuities within the interval of integration is call improper integrals. Examples of improper integrals with infinite intervals of integration:

Examples of improper integrals with infinite discontinuities in the interval of integration:

Examples of improper integrals with infinite discontinuities and infinite intervals of integration:

Definition: The improper integral of f over the interval is defined as:

If lim exist convergeIf lim do not exist diverge

Example 2.3.1

Evaluate a)

b)

c)

Definition: The improper integral of f over the interval is defined as:

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Definition: The improper integral of f over the interval is defined as:

The improper integral is said to converge if both term converge and diverge if either term diverges.

Example 2.3.2

Evaluate using the definition of improper integrals by choosing c = 0.

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