Ap Calculua Ab Limits and Continuity Powerpoint Presentation

Limits and Continuity - PowerPoint PPT Presentation

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Limits and Continuity

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Limits and Continuity Definition Evaluation of Limits Continuity Limits Involving Infinity Examples Find the limits Limit and Trig Functions From the graph of trigs ... – PowerPoint PPT presentation

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Title: Limits and Continuity

1
Limits and Continuity

• Definition

• Evaluation of Limits

• Continuity

• Limits Involving Infinity

2
Limit
L
a
3
Limits, Graphs, and Calculators
Graph 1
Graph 2
4
Graph 3
5
c) Find
6
Note f (-2) 1 is not involved

• 2

6
3) Use your calculator to evaluate the
limits
Answer 16
Answer no limit
Answer no limit
Answer 1/2
7
The Definition of Limit
L
a
8
(No Transcript)
9
Examples
What do we do with the x?
10
1/2
1
3/2
11
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
12
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
13
Examples of One-Sided Limit
Examples
1. Given
Find
Find
14
More Examples
Find the limits
15
A Theorem
This theorem is used to show a limit does not
exist.
For the function
But
16
Limit Theorems
17
Examples Using Limit Rule
Ex.
Ex.
18
More Examples
19
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
20
More Examples
21
The Squeezing Theorem
See Graph
22
Continuity

A function f is continuous at the point x a if
the following are true
f(a)
a
23
A function f is continuous at the point x a if
the following are true
f(a)
a
24
Examples
At which value(s) of x is the given function
discontinuous?
Continuous everywhere
Continuous everywhere except at
25
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
26
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.
27
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
28
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.
29
Limits at Infinity
For all n gt 0,
provided that is defined.
Divide by
Ex.
30
More Examples
31
(No Transcript)
32
(No Transcript)
33
Infinite Limits
For all n gt 0,
More Graphs
34
Examples
Find the limits
35
Limit and Trig Functions
From the graph of trigs functions
we conclude that they are continuous everywhere
36
Tangent and Secant
Tangent and secant are continuous everywhere in
their domain, which is the set of all real
numbers
37
Examples
38
Limit and Exponential Functions
The above graph confirm that exponential
functions are continuous everywhere.
39
Asymptotes
40
Examples
Find the asymptotes of the graphs of the functions
41
(No Transcript)

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