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2.6 – Limits Involving Infinity

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Definition

The notation

lim ( )x a

f x

means that the values of f (x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a (on either side) but not equal to a.

a

f

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Vertical Asymptote

The line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following six statements is true:

lim ( ) ______x a

f x

lim ( ) ______x a

f x

lim ( ) _______x a

f x

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Examples

Find the limit.

1.

2. State all vertical asymptotes for the following function and write the equivalent limit statement for each asymptote.

35lim

( 5)

x

x

e

x

2

2 5( )

4 3

xf x

x x

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Definition

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

lim ( )x

f x L

means that the value of f (x) can be made as close to L as we like by taking x sufficiently large.

L

f

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Horizontal Asymptote

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

lim ( )x

f x L

lim ( )x

f x L

or

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Examples

4. lim x

xe

01. lim ln

xx

Evaluate the following. State the equations of any asymptotes that result from the limit.

2

2. lim tanx

x

15. lim tanx

x

33. lim

( 5)

x

x

e

x

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Algebra Review

1. Simplify

2. Bring the expression into the radical and simplify.

34

3 3

1/2

3 1 1/

xx

x x x

331/ 3 2y y y

n n na b ab

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Properties

lim 0nx

a

x If n is a positive integer, then ,

where a is some constant.

To evaluate limits going to infinity, we often use the technique of multiplying the expression by 1 in the form of

1/

1/

n

n

x

x

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Examples

Evaluate the limit and determine any asymptotes.

1. 2.

3.

2

2

2lim

3 1t

t

t t

2

2lim

9 1y

y

y

2

2

sinlimx

x

x

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You Try It

Evaluate the limit and use the results to state any asymptotes that exist.

1. 2.

3. 4.

3

5

7 3 2lim

4 3p

p p

p

2 2lim y

ye

y

2

1lim

25x

x

x x

2

3lim

2x

x

x