Limits at Infinity, Asymptotes and Dominant terms Presentation Limits... · 12. Computer explorations . 13. ... approaches infinity (written as . x →∞, or . x ... Consider the

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Limits at Infinity, Asymptotes and Dominant terms


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


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.


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.


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.


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→−∞

= −∞ .


2. Basic example: limits at infinity of 1( ) =f xx

This function is defined for all 0x ≠ . We have: 1lim 0

x x→∞= ,

1lim 0x x→−∞

= Proof.

1 1( ) 0f x Ax 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 1xε

> .

As x →∞ it is enough to take any 1Mε

> . The second limit can be proved analogically for


x →−∞ by taking 1Mε

< − .

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 =

0100000 50000 50000 100000

0.0001

0.00005

0.00005

0.0001

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


3. Limits laws as x →±∞

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

xf x A

→±∞= and lim ( )

xg x B

→±∞= . Then

1. Constant multiple rule: { }lim ( )x

f x Aλ 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 Ag x B , 0, 0g B≠ ≠

5. Comparison rule: If ( ) ( ) ( )f x h x g x≤ ≤ , then the limit lim ( )

xh x

→±∞ exists and lim ( )x

A h x B→±∞

≤ ≤ .


4. Examples using limits laws at ±∞ Find the limits:

a) 3

1 2lim 10x x x→∞

− − , b)

3

3

5 6lim3 4x

x xx→∞

−− + , c)

2 1lim→−∞

+x

xx ,

d) sinlim

→∞x

xx , e)

32 1lim1→∞

+−x

xx

Solution a). 3 3

1 2 1 2lim 10 lim lim lim10→∞ →∞ →∞ →∞

− − = − − x x x xx x x x

331 1lim 2lim lim10 0 2.0 10 10

→∞ →∞ →∞

= − − = − − = − x x xx x .


Solution b).

33 3

33

3

6(5 )5 6lim lim 43 4 ( 3 )→∞ →∞

−−=

− + − +x x

xxx x xx x

x

3 3

3 3

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

→∞

→∞

→∞

− − −= = = −

− +− + − +

x

x

x

xx x

x x

Solution c).

22 2

111lim lim→−∞ →−∞

++=

x x

xx xx x

2 2

2

1 11 ( ) 1 1lim lim lim 1 1→−∞ →−∞ →−∞

+ − += = = − + = −

x x x

x xx x

x x x


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

1 sin 1− ≤ ≤x .

Now from the comparison rule:

1 sin 1lim lim lim→∞ →∞ →∞

− ≤ ≤x x x

xx x x , or

sin0 lim 0→∞

− ≤ ≤x

xx .

Therefore sinlim 0.

→∞=

x

xx


Solution e). 3 32 1 2 1lim lim

1 11→∞ →∞

+ +=

− −

x x

x xx

xx

( )22 11 2lim lim2lim

1 11 1 lim

→∞ →∞

→∞

→∞

++ = =

− −

x x

x

x

x xx x x xx

xx x

( )2

22 lim 0

2 lim1 0

→∞

→∞

+= = = ∞

−x

x

x xx x .


5. Remarkable limits at ±∞

1lim 1x

xe

x→±∞

+ = , lim 1

xk

x

k ex→∞

+ =

Examples. Find the limits:

a) 1lim 12→∞

+

x

x x , b)

3lim

1→∞

+

x

x

xx

Solution a). 121 1 1lim 1 lim 1 .

2 2→∞ →∞

+ = + = =

x x

x xe e

x x .


Solution b).

33 3

33 3

1 1 1lim lim 11 11 lim 1

−

→∞ →∞

→∞

= = = = + + +

xx x

xx x

x

x ex e

x x.

33 3

33 3

1 1 1lim lim 11 11 lim 1

−

→∞ →∞

→∞

= = = = + + +

xx x

xx x

x

x ex e

x x


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

0 δ< − <x a ⇒ ( ) >f x L .


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

0 δ< − <x a ⇒ ( ) < −f x L .

Remark. Remember, that the definitions 4-5 do not represent usual limits, these are only notations!


The definitions 4-5 are also used in mathematics for

one-sided infinite limits to a finite number a. It is to express the behavior of the function for all x, situated only 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 af x

+→= ∞ , lim ( )

x af x

−→= ∞ , lim ( )

x af x

+→= −∞ ,

or lim ( )x a

f x−→

= −∞ .


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:

0

1lim+→

= ∞x x .

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

):

0

1lim−→

= −∞x x . See also Fig. 1.


Examples. Find the behavior of the functions near a.

a) 2 , 9

9=

−x a

x , b) 2 1 , 1

1−

=−

x ax .

Solution a). We observe, that at x=9

the denominator becomes 0. We compute the limit above 9:

9

2 18lim9 0+ +

→

= = ∞ − x

xx .

The limit below 9 is:

9

2 18lim9 0− −

→

= = −∞ − x

xx

20 10 10 20 30 40

4

2

2

4

6

8

Fig. 2 Consider the graphics of the function

2( )9

=−xf x

x near x = 9. For x > 9, f(x)→ + ∞;

for x < 0, f (x)→ - ∞.


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 1lim lim1 1

1 1lim 2 lim1 1

+ +

+ +

→ →

→ →

− − +=

− −

+= = = ∞

− −

x x

x x

x x xx x

xx x

.

f (x)

110 5 5 10

1

1

2

3

4

Fig. 3 Graphics of the function

2 1( )1−

=−

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


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


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:

1lim 0→∞

= x x and

1lim 0→−∞

= x x .

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


10. Vertical asymptote

Definition 7. A line x = a is a vertical asymptote of the graph of a function y = f (x) if either lim ( )

+→= ±∞

x af x or lim ( )

−→= ±∞

x af x .

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

0

1lim+→

= +∞ x x and 0

1lim−→

= −∞ x x

which means that the line x = 0 is a vertical asymptote of the function 1/x on both above and below the zero.


Example. Find the horizontal and vertical asymptotes

of the function: 2

23 5 1( )

4− −

=−

x xf xx .

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

point x = ∞ we try to cancel the bigger term (here 2x ): 2

2 2 2

22

2

5 13 13 5 1lim lim 3

44 1→∞ →∞

− − − − = =− −

x x

xxx x x x

x xx


22 2 2

22

2

5 13 13 5 1lim lim 3

44 1→−∞ →−∞

− − − − = =− −

x x

xxx x x x

x xx

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

22 2 2

3 5 1 3.4 5.2 1 1lim lim lim4 ( 2)( 2) 4( 2)+ + +→ → →

− − − −= = = +∞

− − + −x x x

x xx x x x 2

22 2 2

3 5 1 3.4 5.2 1 1lim lim lim4 ( 2)( 2) 4( 2)− − −→ → →

− − − −= = = −∞

− − + −x x x

x xx x x x


2

22 2 2

3 5 1 3.4 5.( 2) 1 21lim lim lim4 ( 2)( 2) 4( 2)+ + +→− →− →−

− − − − −= = = −∞

− − + − +x x x

x xx x x x 2

22 2 2

3 5 1 3.4 5.( 2) 1 21lim lim lim4 ( 2)( 2) 4( 2)− − −→− →− →−

− − − − −= = = +∞

− − + − +x x x

x xx 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.


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 = -2x

y

10 5 5 10

4

2

2

4

6

8

10

Fig. 4 Graphics of the function 2

23 5 1( )

4− −

=−

x xf xx with

its asymptotes (in blue color).


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 xkx and ( )lim ( )

→±∞= −

xb f x kx

if these limits exist.

Example. Find the asymptotes of the function 2 9( )

2−

=−

xf xx .

Solution: Horizontal asymptotes do not exist because:


29 9( ) ( )9lim lim lim2 22 (1 ) (1 )→∞ →∞ →∞

− −−= =

− − −x x x

x x xx x xx x

x x

( 9 / )lim lim1→∞ →∞

−= = = ∞

x x

x x x

and 2 9lim ... lim

2→−∞ →−∞

−= = = −∞

−x x

x xx .

Vertical asymptotes. At x = 2 the function is undefined, but:

2

2 2 2

9 4 9 1lim lim 5 lim2 ( 2) 4( 2)+ + +→ → →

− −= = − = −∞

− − −x x x

xx x x


2

2 2 2

9 4 9 1lim lim 5 lim2 ( 2) 4( 2)− − −→ → →

− −= = − = +∞

− − −x x x

xx 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 xkx and ( )lim ( )

→±∞= −

xb f x kx .

2 2

2

9 (1 9 / )lim lim 1( 2) (1 2 / )→±∞ →∞

− −= =

− −x x

x x xx x x x ⇒ k = 1 at x = ±∞ .

( )2 2 29 9 2lim ( ) lim lim

2 2→±∞ →±∞ →±∞

− − − += − = − = − − x x x

x x x xb f x kx xx x

2 9lim 22→±∞

−= =

−x

xx ⇒ The oblique asymptote is y = x + 2.


Representation of the graphics of the function

2 9( )

2−

=−

xf xx

and its asymptotes: x = 2 , vertical asymptote; y = x + 2, oblique asymptote.

The function approaches the asymptotes at x→±∞ and y→±∞.

2f (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−

=−

xf xx

with its asymptotes (in blue color).


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:

http://www.wolframalpha.com/�



To draw a graphics just type: Plot [(x^2-9)/(x-2), {x,-15,15}]


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−

= = + −− −

xf x xx 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 xx approaches x→∞, respectively.

These are called dominant terms of the function.


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)]


References for further reading:

[1] G. B. Thomas, M. D. Weir., J. Hass, F. R. Giordano, Thomas’ Calculus including second-order differential equations, 11 ed., Pearson Addison-Wesley, 2005.

[2] http://www.wolframalpha.com/

http://www.wolframalpha.com/�

Limits at Infinity, Asymptotes and Dominant terms Presentation Limits... · 12. Computer explorations . 13. ... approaches infinity (written as . x →∞, or . x ... Consider the

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