Section 1.3: Evaluating Limits Analytically

Example

Let and . Find: 4 2f 4 5g 2

3 5 7

2

3 5 7

4

15 7

4

8

4

2

2

3 4 7

4

g

f

Direct SubstitutionOne of the easiest and most useful ways to evaluate a limit analytically is direct substitution (substitution and evaluation):

If you can plug c into f(x) and generate a real number answer in the range of f(x), that generally implies that the limit exists (assuming f(x) is continuous at c).

Example: 3

2limxx

32 8

Always check for substitution first.

The slides that follow investigate why Direct Substitution is valid.

Properties of LimitsLet b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

Constant Function

Limit of x

Limit of a Power of x

Scalar Multiple

lim ( )x cf x L

lim ( )

x cg x K

limx cb b

limx cx c

lim n n

x cx c

lim ( )x c

b f x b L

Properties of LimitsLet b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:

Sum Difference

Product

Quotient

Power

lim ( )x cf x L

lim ( )

x cg x K

lim ( ) ( )x c

f x g x L K

lim ( ) ( )x c

f x g x L K

( )lim , 0

( )x c

f x LK

g x K

lim ( )n n

x cf x L

ExampleLet and . Find the following limits.

5lim 4xf x

5lim 2xg x

1. limx 5

f x 5g x

limx 5f x +lim

x 5 5g x

limx 5

f x +5limx 5

g x

2. limx 5

f x g x

3. limx 5

f x g x

4 +5 2

6

limx 5f x lim

x 5 g x

4 2

8

limx 5

f x

limx 5

g x

4 2

2

Example 2

5 3 2

2Evaluate lim 2 9 3 11

xx x x

5 3 2

2 2 2 2lim 2 lim 9 lim 3 lim 11x x x x

x x x

Sum/Difference Property

5 3 2

2 2 22 lim 9 lim 3 lim 11

x x xx x x

Multiple and Constant Properties5 3 2

2 2 22 lim 9 lim 3 lim 11

x x xx x x

Power Property

5 3 22 2 9 2 3 2 11

Limit of x Property

7

Dire

ct S

ubst

itutio

n

Direct substitution is a valid analytical method to evaluate the following limits.

• 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, then

• If a radical function where n is a positive integer. The following limit is valid for all c if n is odd and only c>0 when n is even:

Direct Substitution

lim ( ) ( )x cp x p c

( )lim ( ) ( ) , ( ) 0

( )x c

p cr x r c q c

q c

lim n n

x cx c

Direct substitution is a valid analytical method to evaluate the following limits.

• If the f and g are functions such that Then the limit of the composition is:

• If c is a real number in the domain of a trigonometric function then:

Direct Substitution

lim ( ) lim ( ) ( )x c x Lg x L and f x f L

lim ( ( )) lim ( ) ( )x c x cf g x f g x f L

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

Example

3

2Evaluate lim

6x

x

x

3 2

3 6

Direct Substitution can be used since the

function is well defined at x=3

1

9

For what value(s) of x can the limit not be evaluated using direct substitution?

At x=-6 since it makes the denominator 0: 6 6 0

Indeterminate Form

0

0

An example of an indeterminate form because the limit can

not be determined. 1/0 is another example.

Often limits can not be evaluated at a value using Direct Substitution. If this is the case, try to find another function that agrees with the original function except at the point in question. In other words…

How can we simplify: ?2

24 4

2x xx x

Evaluate the limit analytically:2

24 4

22lim x x

x xx

2

2

2 4 2 4

2 2 2

Strategies for Finding LimitsTo find limits analytically, try the following:

1. Direct Substitution (Try this FIRST)

2. If Direct Substitution fails, then rewrite then find a function that is equivalent to the original function except at one point. Then use Direct Substitution. Methods for this include…

• Factoring/Dividing Out Technique• Rationalize Numerator/Denominator• Eliminating Embedded Denominators • Trigonometric Identities• Legal Creativity

Example 1Evaluate the limit analytically:

2

24 4

22lim x x

x xx

2 2

2 12lim x x

x xx

Factor the numerator and denominator

2 2

2 12lim x x

x xx

Cancel common factors

212

lim xxx

2 22 1 Direct substitution

0

At first Direct Substitution fails

because x=2 results in dividing by zero

This function is equivalent to the

original function except at x=2

Example 2Evaluate the limit analytically:

2 222

lim yyy

2 2

2 2

y

y

Rationalize the numerator

2 4

2 2 22lim y

y yy

Cancel common factors

2

2 2 22lim y

y yy

12 22

limyy

Direct substitution1 1

42 2 2

Example 3Evaluate the limit analytically:

1 13

33lim x

xx

33xx

Cancel the denominators of the fractions

in the numerator

3

3 33lim x

x xx

If the subtraction is

backwards, Factoring a negative 1 to flip the signs

3

3 33lim x

x xx

3

3 33lim x

x xx

Direct substitution1 13 3 9

Cancel common factors

133

lim xx

limh 0

h h 10 h

Example 4Evaluate the limit analytically:

Expand the the expression

to see if anything cancels

Direct substitution

Factor to see if anything cancels

limh 0

h 5 2 25h

limh 0

h 5 h 5 25h

limh 0

h 2 10h25 25h

limh 0

h 2 10hh

10

limh 0

h 10

0 10

limx 4

sin x cosx sin x cosx cosx

Example 5Evaluate the limit analytically:

Rewrite the tangent

function using cosine and

sine

Direct substitution

Eliminate the embedded fraction

limx 4

1 tan xsin x cosx

limx 4

1 sin xcos x

sin x cosx

cosxcosx

limx 4

cosx sin xsin x cosx cosx

12 2

limx 4

1cosx

1cos

4

If the subtraction is backwards,

Factoring a negative 1 to flip

the signs

2

Two “Freebie” Limits

0

sinlim 1x

x

x

0

1 coslim 0x

x

x

The following limits can be assumed to be true (they will be proven later in the year) to assist in finding other limits:

Use the identities to help with these limits. They are located on the first page of your textbook.

Example Evaluate the limit analytically:

sin350

lim xxx

33

If 3x is the input of the

sine function then 3x needs

to be in the denominator

3sin35 30

lim xxx

3 sin35 30

lim xxx

3 sin35 30

lim xxx

Assumed Trig Limit

35

Scalar Multiple Property

35 1

Isolate the “freebie”

limx 0

sin 2 xx sin x 1cosx

ExampleEvaluate the limit analytically:

Try multiplying by the

reciprocal

A freebie limit and Direct substitution

limx 0

1 cosxx sin x

limx 0

1cosx cosx cos2 xx sin x 1cosx

1cosx1cosx

limx 0

1 cos2 xx sin x 1cosx

limx 0

sin xx 1cosx

1 11cos0

Use the Trigonometry

Laws

12

limx 0

sin xx lim

x 0

11cosx

Split up the limits