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Limits and Continuity

- Definition

- Evaluation of Limits

- Continuity

- Limits Involving Infinity

Limit

L

a

Limits, Graphs, and Calculators

Graph 1

Graph 2

Graph 3

c) Find

6

Note f (-2) 1 is not involved

- 2

3) Use your calculator to evaluate the

limits

Answer 16

Answer no limit

Answer no limit

Answer 1/2

The Definition of Limit

L

a

(No Transcript)

Examples

What do we do with the x?

1/2

1

3/2

One-Sided Limits

One-Sided Limit

The right-hand limit of f (x), as x approaches a,

equals L

written

if we can make the value f (x) arbitrarily close

to L by taking x to be sufficiently close to the

right of a.

L

a

The left-hand limit of f (x), as x approaches a,

equals M

written

if we can make the value f (x) arbitrarily close

to L by taking x to be sufficiently close to the

left of a.

M

a

Examples of One-Sided Limit

Examples

1. Given

Find

Find

More Examples

Find the limits

A Theorem

This theorem is used to show a limit does not

exist.

For the function

But

Limit Theorems

Examples Using Limit Rule

Ex.

Ex.

More Examples

Indeterminate Forms

Indeterminate forms occur when substitution in

the limit results in 0/0. In such cases either

factor or rationalize the expressions.

Notice form

Ex.

Factor and cancel common factors

More Examples

The Squeezing Theorem

See Graph

Continuity

A function f is continuous at the point x a if

the following are true

f(a)

a

A function f is continuous at the point x a if

the following are true

f(a)

a

Examples

At which value(s) of x is the given function

discontinuous?

Continuous everywhere

Continuous everywhere except at

and

and

Thus F is not cont. at

Thus h is not cont. at x1.

F is continuous everywhere else

h is continuous everywhere else

Continuous Functions

If f and g are continuous at x a, then

A polynomial function y P(x) is continuous at

every point x.

A rational function is

continuous at every point x in its domain.

Intermediate Value Theorem

If f is a continuous function on a closed

interval a, b and L is any number between f (a)

and f (b), then there is at least one number c in

a, b such that f(c) L.

f (b)

L

f (c)

f (a)

a

b

c

Example

f (x) is continuous (polynomial) and since f (1)

lt 0 and f (2) gt 0, by the Intermediate Value

Theorem there exists a c on 1, 2 such that f

(c) 0.

Limits at Infinity

For all n gt 0,

provided that is defined.

Divide by

Ex.

More Examples

(No Transcript)

(No Transcript)

Infinite Limits

For all n gt 0,

More Graphs

Examples

Find the limits

Limit and Trig Functions

From the graph of trigs functions

we conclude that they are continuous everywhere

Tangent and Secant

Tangent and secant are continuous everywhere in

their domain, which is the set of all real

numbers

Examples

Limit and Exponential Functions

The above graph confirm that exponential

functions are continuous everywhere.

Asymptotes

Examples

Find the asymptotes of the graphs of the functions

(No Transcript)

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