[PPT] - Limits at Infinity, Asymptotes and Dominant terms 1/38 PowerPoint Presentation

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BUILDING UP VIRTUAL MATHEMATICS LABORATORY

Partnership project LLP-2009-LEO-МP-09, MP 09-05414

Limits at Infinity, Asymptotes and Dominant terms

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Overview

1. Limits as x →±∞
2. Basic example: limits at infinity of f(x)=1/x
3. Limits laws as x →±∞
4. Examples using limits laws at ±∞
5. Remarkable limits at ±∞
6. Infinite limits at x→a
7. Examples on infinite limits at x→a
8. Asymptotes of the graph
9. Horizontal asymptote
10. Vertical asymptote
11. Oblique asymptote
12. Computer explorations
13. Dominant terms

References

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1. Limits as x →±∞

In mathematics, the symbol for infinity In this lesson we will consider functions defined on unbounded intervals like (−∞, a], [a, ∞) or (−∞,∞). is indicated as ∞. It is not a real number. When use ∞ or +∞, this means that the considered values become increasingly large positive numbers. When use −∞, this means that the values become decreasingly large negative numbers.

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By analogy with functions on finite intervals it is possible that the function values are bounded when the argument x approaches infinity (written as x →∞, or x → −∞, or x →±∞). In many cases the function values can approach a finite number, called limit. Definition 1. A function f (x) has the limit A as x approaches infinity, noted by

lim ( )

x

f x A

→∞

=

if, for every number ε > 0, there exists a corresponding number M such that for all x > M follows | f (x) − A| < ε.

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Definition 2. A function f (x) has the limit A as x approaches minus infinity, noted by

lim ( )

x

f x A

→−∞

=

if, for every number ε > 0, there exists a corresponding number M such that for all x < M follows | f (x) − A| < ε.

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Definition 3. If for a function f (x) no limit exists as x approaches +∞ or −∞, but all corresponding values increase (decrease) infinitely to +∞ (or −∞) we will say formally that the function limit is +∞ (or −∞), call it infinite limit and denote as

lim ( )

x

f x

→∞

= ∞ , lim ( )

x

f x

→∞

= −∞ , lim ( )

x

f x

→−∞

= ∞ , lim ( )

x

f x

→−∞

= −∞ .

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2. Basic example: limits at infinity of

1 ( ) = f x x

This function is defined for all

x ≠

. We have:

1 lim

x

→∞

= , 1 lim

x

→−∞

=

Proof.

1 1 ( ) f x A x x ε − = − = <

According to the Definition 1, we fix some ε > 0 and we seek for a corresponding M such that for A = 0 and all x > M we will have , from where

1 x ε >

. As x → ∞ it is enough to take any

1 M ε >

. The second limit can be proved analogically for

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x → −∞ by taking

1 M ε < − .

The behavior of the function is given in Fig.1. It shows that:

f (x) decreases to 0

when x→∞ with positive values

f (x) increases to 0

when x→−∞ with negative values.

f (x )=1/x lim = 0 lim = 0 lim = -  lim =  100000 50000 50000 100000 0.0001 0.00005 0.00005 0.0001

Fig. 1 Graphics of f (x) =1/x .

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3. Limits laws as x →±∞

Let A, B and λ are real numbers and there exist the limits: lim

( )

x

f x A

→±∞

=

and lim

( )

x

g x B

→±∞

=

. Then

1. Constant multiple rule:

{ }

lim ( )

x

f x A λ λ

→±∞

=

2. Sum/difference rule:

{ }

lim ( ) ( )

x

f x g x A B

→±∞

± = ±

3. Product rule:

{ }

lim ( ). ( ) .

x

f x g x A B

→±∞

=

4. Quotient rule:

( ) lim ( )

→±∞

=

x

f x A g x B ,

0, g B ≠ ≠

5. Comparison rule:

If

( ) ( ) ( ) f x h x g x ≤ ≤

, then the limit lim

( )

x

h x

→±∞

exists and

lim ( )

x

A h x B

→±∞

≤ ≤

.

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4. Examples using limits laws at ±∞

Find the limits: a)

3

1 2 lim 10

x

x x

→∞

  − −    , b)

3 3

5 6 lim 3 4

x

x x x

→∞

− − + , c)

2

1 lim

→−∞

+

x

x x

, d)

sin lim

→∞ x

x x , e)

3

2 1 lim 1

→∞

+ −

x

x x

Solution a).

3 3

1 2 1 2 lim 10 lim lim lim10

→∞ →∞ →∞ →∞

  − − = − −    

x x x x

3 3

1 1 lim 2lim lim10 2.0 10 10

→∞ →∞ →∞

  = − − = − − = −    

x x x

x x

.

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Solution b).

3 3 3 3 3 3

6 (5 ) 5 6 lim lim 4 3 4 ( 3 )

→∞ →∞

− − = − + − +

x x

x x x x x x x x

3 3 3 3

6 1 (5 ) (5 6lim ) (5 0) 5 lim 4 1 ( 3 0) 3 ( 3 ) ( 3 4lim )

→∞ →∞ →∞

− − − = = = − − + − + − +

x x x

x x x x x

Solution c).

2 2 2

1 1 1 lim lim

→−∞ →−∞

+ + =

x x

x x x x x

2 2 2

1 1 1 ( ) 1 1 lim lim lim 1 1

→−∞ →−∞ →−∞

+ − + = = = − + = −

x x x

x x x x x x x

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Solution d). We know that for all real x:

1 sin 1 − ≤ ≤ x

. Now from the comparison rule:

1 sin 1 lim lim lim

→∞ →∞ →∞

− ≤ ≤

x x x

x x x x , or sin lim

→∞

− ≤ ≤

x

x x

. Therefore

sin lim 0.

→∞

=

x

x x

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Solution e).

3 3

2 1 2 1 lim lim 1 1 1

→∞ →∞

+ + = −   −    

x x

x x x x x

( )

2 2

1 1 2lim lim 2 lim 1 1 1 1 lim

→∞ →∞ →∞ →∞

    + +         = = − −

x x x x

x x x x x x x x x x

( )

2 2

2lim 2lim 1 0

→∞ →∞

+ = = = ∞ −

x x

x x x x

.

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5. Remarkable limits at ±∞

1 lim 1

x x

e x

→±∞

  + =    

, lim

1 x k x

k e x

→∞

  + =    

Examples. Find the limits:

a)

1 lim 1 2

→∞

  +    

x x

x

, b)

3 lim 1

→∞

    +  

x x

Solution a).

1 2

1 1 1 lim 1 lim 1 . 2 2

→∞ →∞

    + = + = =        

x x x x

e e x x

.

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Solution b).

3 3 3 3 3 3

1 1 1 lim lim 1 1 1 1 lim 1

− →∞ →∞ →∞

      = = = =     +       + +      

x x x x x x x

x e x e x x

.

3 3 3 3 3 3

1 1 1 lim lim 1 1 1 1 lim 1

− →∞ →∞ →∞

      = = = =     +       + +      

x x x x x x x

x e x e x x

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6. Infinite limits at x→a

In many cases a function can grow or decrease infinitely when x approaches a finite number a. In fact this shows the behavior of the functions near a. Definition 4. We say that f (x) approaches infinity as x approaches a, and note lim

( )

→

= ∞

x a f x

If for every positive real number L there exist a corresponding number δ > 0 such that for all x satisfying

δ < − < x a

⇒

( ) > f x L .

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Definition 5. We say that f (x) approaches minus infinity as x approaches a, and note lim

( )

→

= −∞

x a f x

If for every positive real number L there exist a corresponding number δ > 0 such that for all x satisfying

δ < − < x a

⇒

( ) < − f x L .

Remark. Remember, that the definitions 4-5 do not

represent usual limits, these are only notations!

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The definitions 4-5 are also used in mathematics for

ne-sided infinite limits to a finite number a. It is to

express the behavior of the function for all x, situated

nly at the left side of a (denoted as x → a+) or to the

right side of a (denoted as x → a− ). For instance: lim

( )

x a

f x

+

→

= ∞, lim ( )

x a

f x

−

→

= ∞, lim ( )

x a

f x

+

→

= −∞,

r lim

( )

x a

f x

−

→

= −∞ .

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7. Examples on infinite limits at x→a

Basic example. The function f (x)=1/x is not defined at x = 0. But for all positive x very closed to 0, denoted as x → 0+ , the function increases infinitely and surpasses every positive real number. This is the meaning of definition 4 for the left side. Therefore:

1 lim

+

→

= ∞

x

. Respectively, for all negative x near 0 (say x → 0− ):

1 lim

−

→

= −∞

x

. See also Fig. 1.

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Examples. Find the behavior of the functions near a.

a)

2 , 9 9 = − x a x

, b)

2 1 , 1 1 − = − x a x

. Solution a).

We observe, that at x=9

the denominator becomes

0. We compute the limit

above 9:

9 2 18 lim 9

+

+ →

  = = ∞   −  

x

x x

.

The limit below 9 is:

9 2 18 lim 9

−

− →

  = = −∞   −  

x

x x

20 10 10 20 30 40 4 2 2 4 6 8

Fig. 2 Consider the graphics of the function

2 ( ) 9 = − x f x x

near x = 9. For x > 9, f(x)→ + ∞;

for x < 0, f (x)→ - ∞.

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Solution b).

The domain of definition

is

( , 1] (1, ) = −∞ − ∪ ∞ D

.

The singular point is

1 = = x a

, where the denominator is 0. The limit above a is

2 1 1 1 1

1 1 1 lim lim 1 1 1 1 lim 2 lim 1 1

+ + + +

→ → → →

− − + = − − + = = = ∞ − −

x x x x

x x x x x x x x

.

f (x ) 1 10 5 5 10 1 1 2 3 4

Fig. 3 Graphics of the function

2

1 ( ) 1 − = − x f x x . Near x = 1, x > 1, f (x)→ + ∞.

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8. Asymptotes of the graph

If the distance between the graph of a function and some fixed line approaches zero as a point on the graph moves increasingly far from the origin, we say that the graph approaches the line asymptotically and that the line is an asymptote of the graph . There are three types of asymptotes:

Horizontal asymptotes
Vertical asymptotes
Oblique asymptotes

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9. Horizontal asymptote

Definition 6. A line y = b is a horizontal asymptote of the graph of a function y = f (x) if either

lim ( )

→∞

=

x

f x b or lim

( )

→−∞

=

x

f x b .

Basic example. As we saw in section 2 and Fig. 1:

1 lim

→∞

  =    

x

and

1 lim

→−∞

  =    

x

. This way the line y = 0 is a horizontal asymptote of the function 1/x on both infinity and minus infinity.

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10. Vertical asymptote

Definition 7. A line x = a is a vertical asymptote

f the graph of a function y = f (x) if either

lim ( )

+

→

= ±∞

x a

f x

r lim

( )

−

→

= ±∞

x a

f x .

Basic example. In section 7 (see also Fig. 1) we

btained:

1 lim

+

→

  = +∞    

x

and

1 lim

−

→

  = −∞    

x

which means that the line x = 0 is a vertical asymptote

f the function 1/x on both above and below the zero.

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Example. Find the horizontal and vertical asymptotes
f the function:

2 2

3 5 1 ( ) 4 − − = − x x f x x

. Solution.

Horizontal asymptotes are at x→±∞. For the singular

point x = ∞ we try to cancel the bigger term (here

2 x ):

2 2 2 2 2 2 2

5 1 3 1 3 5 1 lim lim 3 4 4 1

→∞ →∞

  − −   − −   = = −   −    

x x

x x x x x x x x x

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2 2 2 2 2 2 2

5 1 3 1 3 5 1 lim lim 3 4 4 1

→−∞ →−∞

  − −   − −   = = −   −    

x x

x x x x x x x x x

We conclude that the line y = 3 is a horizontal asymptote both at infinity and negative infinity.

Vertical asymptotes are at x = ±2. We compute all

four possibility limits:

2 2 2 2 2

3 5 1 3.4 5.2 1 1 lim lim lim 4 ( 2)( 2) 4( 2)

+ + +

→ → →

− − − − = = = +∞ − − + −

x x x

x x x x x x

2 2 2 2 2

3 5 1 3.4 5.2 1 1 lim lim lim 4 ( 2)( 2) 4( 2)

− − −

→ → →

− − − − = = = −∞ − − + −

x x x

x x x x x x

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2 2 2 2 2

3 5 1 3.4 5.( 2) 1 21 lim lim lim 4 ( 2)( 2) 4( 2)

+ + +

→− →− →−

− − − − − = = = −∞ − − + − +

x x x

x x x x x x

2 2 2 2 2

3 5 1 3.4 5.( 2) 1 21 lim lim lim 4 ( 2)( 2) 4( 2)

− − −

→− →− →−

− − − − − = = = +∞ − − + − +

x x x

x x x x x x

We conclude that the lines x = ±2 are vertical asymptotes both at the two sides. The graphics of the functions and its asymptotes is shown in Fig. 4.

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Conclusion:

f (x) approaches the

value 3 at ±∞.

f (x) approaches ∞

when x approaches −2 from below and +2 from above.

f (x) approaches −∞

when x approaches −2 from above and +2 from below.

2

2

f (x ) y = 3 y = 3 x = 2 x = -2 x y 10 5 5 10 4 2 2 4 6 8 10

Fig. 4 Graphics of the function

2 2

3 5 1 ( ) 4 − − = − x x f x x with its asymptotes (in blue color).

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11. Oblique asymptote

Definition 8. A line y = kx + b is an oblique asymptote of the graph of a function y = f (x) where

( ) lim

→±∞

=

x

f x k x and

( )

lim ( )

→±∞

= −

x

b f x kx if these limits exist.

Example. Find the asymptotes of the function

2

9 ( ) 2 − = − x f x x

. Solution:

Horizontal asymptotes do not exist because:

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2 9 9 ( ) ( ) 9 lim lim lim 2 2 2 (1 ) (1 )

→∞ →∞ →∞

− − − = = − − −

x x x

x x x x x x x x x x ( 9/ ) lim lim 1

→∞ →∞

− = = = ∞

x x

x x x

and

2 9 lim ... lim 2

→−∞ →−∞

− = = = −∞ −

x x

x x x

.

Vertical asymptotes. At x = 2 the function is

undefined, but:

2 2 2 2

9 4 9 1 lim lim 5 lim 2 ( 2) 4( 2)

+ + +

→ → →

− − = = − = −∞ − − −

x x x

x x x x

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2 2 2 2

9 4 9 1 lim lim 5 lim 2 ( 2) 4( 2)

− − −

→ → →

− − = = − = +∞ − − −

x x x

x x x x

. So, the line x = 2 is a two-sided vertical asymptote.

Oblique asymptotes. We try to find y = kx + b where

( ) lim

→±∞

=

x

f x k x and

( )

lim ( )

→±∞

= −

x

b f x kx .

2 2 2

9 (1 9/ ) lim lim 1 ( 2) (1 2/ )

→±∞ →∞

− − = = − −

x x

x x x x x x x

⇒ k = 1 at x = ±∞ .

( )

2 2 2

9 9 2 lim ( ) lim lim 2 2

→±∞ →±∞ →±∞

  − − − + = − = − =     − −  

x x x

x x x x b f x kx x x x 2 9 lim 2 2

→±∞

− = = −

x

x x

⇒ The oblique asymptote is y = x + 2.

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Representation of the graphics of the function

2

9 ( ) 2 − = − x f x x

and its asymptotes: x = 2 , vertical asymptote; y = x + 2, oblique asymptote. The function approaches the asymptotes at x→±∞ and y→±∞.

2 f (x ) x y y = x + 2 x =2 10 5 5 10 20 10 10 20

Fig. 5 Graphics of the function

2

9 ( ) 2 − = − x f x x with its asymptotes (in blue color).

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12. Computer explorations

For simple calculations we can use directly the Mathematica computational knowledge online engine http://www.wolframalpha.com/ For instance to compute the limit of the function at infinity we just type the formula like this: Limit[(x^2-9)/(x-2), x->infinity] The result is as follows:

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To draw a graphics just type:

Plot [(x^2-9)/(x-2), {x,-15,15}]

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13. Dominant terms

In most cases, we may find a representation of a function where one part of the formula expresses the behavior of the function at its singular points.

Example. Let us represent the previous function as

( )

2

9 5 ( ) 2 2 2 − = = + − − − x f x x x x

For x→±∞ the second term vanishes, so

( )

( ) 2 ≈ + f x x

. When x→±2, the first term is fixed and the function

5 ( ) 2 ≈ − − f x x

approaches x→  ∞, respectively. These are called dominant terms of the function.

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The dominant terms can be found by dividing polynomials, by using the series representation etc. By the Wolfram alfa Mathematica engine just type

Apart[(x^2-9)/(x-2)]

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References for further reading: [1] G. B. Thomas, M. D. Weir., J. Hass, F. R. Giordano, Thomas’ Calculus including second-

rder differential equations, 11 ed., Pearson

Addison-Wesley, 2005. [2] http://www.wolframalpha.com/