limits at infinity, asymptotes and dominant terms presentation limits... · 12. computer...
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------------------ Snezhana Gocheva-Ilieva, Plovdiv University --------------------- 1/38
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 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
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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 .
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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→±∞
≤ ≤ .
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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 .
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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
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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
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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 .
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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 .
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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
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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
0 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
0 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
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−→
= −∞ .
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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:
0
1lim+→
= ∞x x .
Respectively, for all negative x near 0 (say x → 0−
):
0
1lim−→
= −∞x 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 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)→ - ∞.
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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 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)→ + ∞.
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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:
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.
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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.
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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
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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
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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.
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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 = -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).
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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 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:
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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
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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.
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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).
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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−
= = + −− −
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.
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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-order differential equations, 11 ed., Pearson Addison-Wesley, 2005.
[2] http://www.wolframalpha.com/