Limits at Infinity Explore the End Behavior of a function.

Limits at Infinity

Explore the End Behavior of a function

End Behavior

• The following is very important.

01

lim xx

01

lim xx

End Behavior

• The following is very important.• When the limit at infinity exists, it is known as the

horizontal asymptote.

Example 2

2tanlim 1

x

x 2tanlim 1

x

x

Example 3

• The following limits are one definition of e.

ex

x

x

11lim e

x

x

x

11lim

Polynomials

• We can generalize the end behavior of polynomials.

Polynomials

• We can generalize the end behavior of polynomials.• If the leading coefficient of the polynomial is positive:

n

xxlim

n

xxlim

,...5,3,1, n

,...6,4,2, n

Polynomials

• We can generalize the end behavior of polynomials.• If the leading coefficient of the polynomial is

negative:

n

xxlim

n

xxlim

,...5,3,1, n

,...6,4,2, n

Polynomials

• The end behavior of a polynomial matches the end behavior of its highest degree term.

Polynomials

• The end behavior of a polynomial matches the end behavior of its highest degree term.

• For example:

5lim xx

162lim 45 xxxx

Limits of Rational Functions

• We will have three different situations.

Limits of Rational Functions

• We will have three different situations.1. The degree of the numerator and the denominator is

the same.

Limits of Rational Functions

• We will have three different situations.1. The degree of the numerator and the denominator is

the same.2. The degree of the denominator is higher than the

degree of the numerator.

Limits of Rational Functions

• We will have three different situations.1. The degree of the numerator and the denominator is

the same.2. The degree of the denominator is higher than the

degree of the numerator.3. The degree of the numerator is higher than the

degree of the denominator.

Same Degree

• Find:86

53lim

x

xx

Same Degree

• Find:

• We will divide the numerator and denominator by the highest power of x in the denominator.

86

53lim

x

xx

x

xx 8

6

53

lim

Same Degree

• Find:

• We will divide the numerator and denominator by the highest power of x in the denominator.

86

53lim

x

xx

2

1

6

3lim

86

53

lim

xx

x

x

Same Degree

• When the numerator and the denominator are the same degree, your answer is a number, and it is the quotient of the leading coefficients.

Denominator is higher power

• We start the same way; divide the numerator and denominator by the highest power of x in the denominator.

• Find:

52

4lim

3

2

x

xxx

Denominator is higher power

• We start the same way; divide the numerator and denominator by the highest power of x in the denominator.

• Find:

33

3

33

2

3

2

52

4

52

4lim

xxx

xx

xx

x

xxx

Denominator is higher power

• We start the same way; divide the numerator and denominator by the highest power of x in the denominator.

• Find:

02

05

2

14

52

4

52

4lim

3

2

33

3

33

2

3

2

x

xx

xxx

xx

xx

x

xxx

Denominator is higher power

• Conclusion: When the denominator has the highest power of x, the answer is zero.

• For fractions, if the denominator accelerates faster than the numerator, the limit is zero.

Numerator is higher power

• We start the same way; divide the numerator and denominator by the highest power of x in the denominator.

• Find:

x

xxx 31

125lim

23

Numerator is higher power

• We start the same way; divide the numerator and denominator by the highest power of x in the denominator.

• Find:

xx

x

xxx

xx

x

xxx 31

125

31

125lim

23

23

Numerator is higher power

• We start the same way; divide the numerator and denominator by the highest power of x in the denominator.

• Find:

3

2531

125

31

125lim

2

23

23

xx

xx

x

xxx

xx

x

xxx

Numerator is higher power

• Conclusion: When the numerator is the higher power of x, the answer is .

• In the past example, the function reacts like a negative second degree equation, so the answer is

.

3

5

3

25lim

22 xxxx

Limits involving radicals

• Find:3

86

53lim

x

xx

Limits involving radicals

• Find:

3

86

53lim

x

xx

3

86

53lim

x

xx

Limits involving radicals

• Find:

33

2

1

86

53lim

x

xx

3

86

53lim

x

xx

Limits involving radicals

• Find:

63

2lim

2

x

xx

Limits involving radicals

• Find:

63

2lim

2

x

xx

||2 xx

||6

||3

2

lim22

2

xxx

xxx

x

Limits involving radicals

• Find:

• If x is approaching positive infinity, the absolute value of x is just x.

63

2lim

2

x

xx

||2 xx

xxxx

xxx

xxx

x 63

21

||6

||3

2

lim222

2

Limits involving radicals

• Find:

63

2lim

2

x

xx

||2 xx

3

16

3

21

||6

||3

2

lim222

2

x

x

xxx

xxx

x

Limits involving radicals

• Find:63

2lim

2

x

xx

||2 xx

Limits involving radicals

• Find:63

2lim

2

x

xx

||6

||3

2

lim22

2

xxx

xxx

x

||2 xx

Limits involving radicals

• Find:

• If x is approaching negative infinity, the absolute value of x is the opposite of x.

63

2lim

2

x

xx

3

163

21

||6

||3

2

lim222

2

xxx

x

xxx

xxx

x

||2 xx

Limits involving radicals

• Find: )5(lim 36 xxx

Limits involving radicals

• Find: )5(lim 36 xxx

3636

3636

5

5lim

)5(

)5()5(lim

xxxx

xxxx

xx

Limits involving radicals

• Find: )5(lim 36 xxx

3636

3636

5

5lim

)5(

)5()5(lim

xxxx

xxxx

xx

0101

0lim

3

3

66

6

3

5

5

xx

xxx

x

x

Limits involving radicals

• Find: )5(lim 336 xxxx

Limits involving radicals

• Find: )5(lim 336 xxxx

336

3

336

336336

5

5lim

)5(

)5()5(lim

xxx

x

xxx

xxxxxx

xx

Limits involving radicals

• Find: )5(lim 336 xxxx

336

3

336

336336

5

5lim

)5(

)5()5(lim

xxx

x

xxx

xxxxxx

xx

2

5

101

5lim

3

3

6

3

6

6

3

3

5

5

xx

xx

xx

xx

x

Limits of exponential and logarithmic functions

xx

lnlim x

xlnlim

0

Limits of exponential and logarithmic functions

• Let’s look at ex.

x

xelim 0lim

x

xe

Limits of exponential and logarithmic functions

• Let’s look at e-x

0lim

x

xe

x

xelim

Homework

• Section 1.3• Pages 96-97• 1-31odd