/
1.3 Evaluating Limits Analytically 1.3 Evaluating Limits Analytically

1.3 Evaluating Limits Analytically - PowerPoint Presentation

pasty-toler
pasty-toler . @pasty-toler
Follow
416 views
Uploaded On 2015-11-29

1.3 Evaluating Limits Analytically - PPT Presentation

Objective Evaluate a limit using properties of limits Miss Battaglia ABBC Calculus Properties of Limits Remember that the limit of fx as x approaches c does not depend on the value of f at xc But it might happen ID: 208813

theorem limit function limits limit theorem limits function real find functions rational properties number trig polynomial positive exists integer

Share:

Link:

Embed:

Download Presentation from below link

Download Presentation The PPT/PDF document "1.3 Evaluating Limits Analytically" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

Slide1

1.3 Evaluating Limits AnalyticallyObjective: Evaluate a limit using properties of limits

Miss

Battaglia

AB/BC CalculusSlide2

Properties of Limits

Remember that the limit of f(x) as x approaches c does not depend on the value of f at x=c… But it might happen!

Direct substitution

substitute x for cThese functions are continuous at cSlide3

Properties of Limits

Theorem 1.1 Some Basic Limits

Let b and c be real numbers and let n be a positive integer.

1. 2. 3.

Example: Evaluating Basic Limits

a) b) c)Slide4

Theorem 1.2: Properties of Limits

Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits.

and

Scalar Multiple:Sum or difference:

Product:Quotient:Power:Slide5

The Limit of a PolynomialSlide6

Theorem 1.3: Limits of Polynomial and Rational Functions

If p is a polynomial function and c is a real number, then

If r is a rational function given by r(x)=p(x)/q(x) and c is a real number such that q(c)≠0, then Slide7

The Limit of a Rational Function

Find the limit:Slide8

Theorem 1.4: The Limit of a Rational Function

Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c>0 if n is even.

Theorem 1.5: The Limit of a Composite Function

If f and g are functions such that

and , thenSlide9

Limit of a Composite Function

f(x)=x

2

+ 4 and g(x)=FindFind

FindSlide10

Theorem 1.6: Limits of Trigonometric Functions

Let c be a real number in the domain of the given trigonometric function.

1. 2.

3. 4.5. 6.

Examples:

a.

b.

c.Slide11

Theorem 1.7: Functions that Agree at All But One Point

Let c be a real number and let f(x)=g(x) for all

x≠c

in an open interval containing c. If the limit of g(x) as x approaches c exists, then the limit of f(x) also exists and

Find the limit:Slide12

Dividing Out Technique

Find the limit:Slide13

Rationalizing Technique

Find the limit:Slide14

Theorem 1.8: The Squeeze Theorem

If h(x)

<

f(x) < g(x) for all x in an open interval containing c,

except possibly at c itself, and ifthen exists and is equal to L.

Theorem 1.9: Two Special Trig Limits

1. 2.Slide15

Extra ExampleSlide16

A Limit Involving a Trig Function

Find the limit:Slide17

A Limit Involving a Trig Function

Find the limit:Slide18

Read 1.3Page 67 #17-75 every other odd, 85-89, 107, 108, 117-122

Classwork/Homework