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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.

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limits and continuity
Limits and Continuity
  • Definition
  • Evaluation of Limits
  • Continuity
  • Limits Involving Infinity
slide5

c) Find

6

Note: f (-2) = 1

is not involved

  • 2
slide6

3) Use your calculator to evaluate the limits

Answer : 16

Answer : no limit

Answer : no limit

Answer : 1/2

examples
Examples

What do we do with the x?

slide10

1/2

1

3/2

one sided limits
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

slide12

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 of One-Sided Limit

Examples

1. Given

Find

Find

more examples
More Examples

Find the limits:

a theorem
A Theorem

This theorem is used to show a limit does not exist.

For the function

But

indeterminate forms
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

continuity
Continuity

A function f is continuous at the point x = a if the following are true:

f(a)

a

examples24

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

Examples

Continuous everywhere

Continuous everywhere except at

slide25

and

and

Thus F is not cont. at

Thus h is not cont. at x=1.

F is continuous everywhere else

h is continuous everywhere else

continuous functions
Continuous Functions

If f and g are continuous at x = a, then

A polynomial functiony = P(x) is continuous at every point x.

A rational function is continuous at every point x in its domain.

intermediate value theorem
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)

f (c) =

L

f (a)

a

c

b

example
Example

f (x) is continuous (polynomial) and since f (1) < 0 and f (2) > 0, by the Intermediate Value Theorem there exists a c on [1, 2] such that f (c) = 0.

limits at infinity
Limits at Infinity

For all n > 0,

provided that is defined.

Divide by

Ex.

infinite limits
Infinite Limits

For all n > 0,

More Graphs

examples34
Examples

Find the limits

limit and trig functions
Limit and Trig Functions

From the graph of trigs functions

we conclude that they are continuous everywhere

tangent and secant
Tangent and Secant

Tangent and secant are continuous everywhere in their domain, which is the set of all real numbers

limit and exponential functions
Limit and Exponential Functions

The above graph confirm that exponential functions are continuous everywhere.

examples40
Examples

Find the asymptotes of the graphs of the functions