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EVALUATING LIMITS ANALYTICALLY
EVALUATING LIMITS ANALYTICALLY
Section 1.3
When you are done with your homework, you should
be able to…• Evaluate a Limit Using Properties of
Limits• Develop and Use a Strategy for Finding
Limits• Evaluate a Limit Using Dividing Out and
Rationalizing Techniques• Evaluate a Limit Using the Squeeze
Theorem
SOME BASIC LIMITSLet and b and c be real numbers
and let n be a positive integer.
1.
2.
3.
limx cb b
limx cx c
lim n n
x cx c
Evaluate
A. 5B. -3C. Does not exist
3lim 5x
Evaluate
A. 1B. -1C. 5D. Does not exist
5
1limx
x
Properties of LimitsLet and b and c be real numbers, let n be a
positive integer, and let f and g be functions with the following limits:
1. Scalar multiple2. Sum or difference3. Product
4. Quotient
5. Power
lim and lim x c x cf x L g x K
limx c
bf x bL
lim
x cf x g x L K
limx c
f x g x LK
lim , provided 0x c
f x LK
g x K
limn n
x cf x L
Evaluate
A. -4B. -2C. 1D. Does not exist
2 3
2lim 5x
x x
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
and c is a real numbersuch that then
limx cp x p c
( )( )
p xq xr x
0,q c
lim .x c
p cr x r c
q c
Evaluate
A. 5B. 0C. -5D. Does not exist
20
5limx
x
x
Evaluate the function at x = 2
A. 1B. DNE
2 2 13 21
2 2
x xx xf x x
x x
The Limit of a Function Involving a Radical
Let n be a positive integer. The following limit is valid for all c if is n odd, and is valid for if n is even.
0c
lim n n
x cx c
The Limit of a Composite Function
• Let f and g be functions with the following limits:
Then
lim and lim x c x Lg x L f x f L
lim limx c x cf g x f g x f L
Limits of Trigonometric Functions
Let c be a real number in the domain of the given trigonometric function.
limsin sinx c
x c
lim cos cosx c
x c
lim tan tanx c
x c
lim cot cotx c
x c
limsec secx c
x c
lim csc cscx c
x c
STRATEGIES FOR FINDING LIMITS
• Functions That Agree at All But One Point– Let c be a real number and let
for all x in an open interval containing c. If the limit of g as x approaches c exists, then the limit of f also exists and .
f x g x
lim limx c x cf x g x
A Strategy for Finding Limits• Learn to recognize which limits can be
evaluated by direct substitution.
• If the limit of f(x) as x approaches c cannot be evaluated by direct substitution, try to find a function g that agrees with f for all x other than x = c.
• Apply .• Use a graph or table to reinforce your
conclusion.
lim limx c x cf x g x g c
Dividing Out Techniques
Example:
22
2lim
4x
x
x
Evaluate.
A. 0 B. Does not existC. 1/16
32
2lim
8x
x
x
Rationalizing Techniques
Example:
0
2 2limx
x
x
Evaluate the exact
limit. 6
2 2lim
6x
x
x
TWO SPECIAL TRIGONOMETRIC LIMITS
0
sinlim 1x
x
x
0
1 coslim 0x
x
x
Evaluate the exact answer.
0
sin 5limx
x
x