lesson 7: limits at infinity

Section 2.5Limits at Infinity

Math 1a

October 10, 2007

Announcements

I Midterm I is coming: October 24, 7:00-9:00 in Halls A and C

Math 1a - October 10, 2007.GWBWednesday, Oct 10, 2007

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DefinitionLet f be a function defined on some interval (a,∞). Then

limx→∞

f (x) = L

means that the values of f (x) can be made as close to L as welike, by taking x sufficiently large.

DefinitionThe line y = L is a called a horizontal asymptote of the curvey = f (x) if either

limx→∞

f (x) = L or limx→−∞

f (x) = L.

y = L is a horizontal line!

DefinitionLet f be a function defined on some interval (a,∞). Then

limx→∞

f (x) = L

means that the values of f (x) can be made as close to L as welike, by taking x sufficiently large.

DefinitionThe line y = L is a called a horizontal asymptote of the curvey = f (x) if either

limx→∞

f (x) = L or limx→−∞

f (x) = L.

y = L is a horizontal line!

DefinitionLet f be a function defined on some interval (a,∞). Then

limx→∞

f (x) = L

means that the values of f (x) can be made as close to L as welike, by taking x sufficiently large.

DefinitionThe line y = L is a called a horizontal asymptote of the curvey = f (x) if either

limx→∞

f (x) = L or limx→−∞

f (x) = L.

y = L is a horizontal line!

TheoremLet n be a positive integer. Then

I limx→∞

1

xn= 0

I limx→−∞

1

xn= 0

Math 1a - October 10, 2007.GWBWednesday, Oct 10, 2007

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Using the limit laws to compute limits at ∞

Example

Find

limx→∞

2x3 + 3x + 1

4x3 + 5x2 + 7

if it exists.

A does not exist

B 1/2

C 0

D ∞

Using the limit laws to compute limits at ∞

Example

Find

limx→∞

2x3 + 3x + 1

4x3 + 5x2 + 7

if it exists.

A does not exist

B 1/2

C 0

D ∞

SolutionFactor out the largest power of x from the numerator anddenominator. We have

2x3 + 3x + 1

4x3 + 5x2 + 7=

x3(2 + 3/x2 + 1/x3)

x3(4 + 5/x + 7/x3)

limx→∞

2x3 + 3x + 1

4x3 + 5x2 + 7= lim

x→∞

2 + 3/x2 + 1/x3

4 + 5/x + 7/x3

=2 + 0 + 0

4 + 0 + 0=

1

2

Upshot

When finding limits of algebraic expressions at infinitely, look atthe highest degree terms.

SolutionFactor out the largest power of x from the numerator anddenominator. We have

2x3 + 3x + 1

4x3 + 5x2 + 7=

x3(2 + 3/x2 + 1/x3)

x3(4 + 5/x + 7/x3)

limx→∞

2x3 + 3x + 1

4x3 + 5x2 + 7= lim

x→∞

2 + 3/x2 + 1/x3

4 + 5/x + 7/x3

=2 + 0 + 0

4 + 0 + 0=

1

2

Upshot

When finding limits of algebraic expressions at infinitely, look atthe highest degree terms.

Another Example

Example

Find

limx→∞

√3x4 + 7

x2 + 3

SolutionThe limit is

√3.

Math 1a - October 10, 2007.GWBWednesday, Oct 10, 2007

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Another Example

Example

Find

limx→∞

√3x4 + 7

x2 + 3

SolutionThe limit is

√3.

Example

Make a conjecture about limx→∞

x2

2x.

SolutionThe limit is zero. Exponential growth is infinitely faster thangeometric growth

Math 1a - October 10, 2007.GWBWednesday, Oct 10, 2007

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Example

Make a conjecture about limx→∞

x2

2x.

SolutionThe limit is zero. Exponential growth is infinitely faster thangeometric growth

Rationalizing to get a limit

Example

Compute limx→∞

(√4x2 + 17− 2x

).

SolutionThis limit is of the form ∞−∞, which we cannot use. So werationalize the numerator (the denominator is 1) to get anexpression that we can use the limit laws on.

Math 1a - October 10, 2007.GWBWednesday, Oct 10, 2007

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Rationalizing to get a limit

Example

Compute limx→∞

(√4x2 + 17− 2x

).

SolutionThis limit is of the form ∞−∞, which we cannot use. So werationalize the numerator (the denominator is 1) to get anexpression that we can use the limit laws on.

Worksheet