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