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
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.
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