ASYMPTOTES
1. Vertical Asymptotes
Definition: A function y = f(x) has the line x = a as a vertical asymptote if
lim f ( x ) = ± ∞ or/and
x→a+
lim f ( x ) = ± ∞
x→a−
(e.g.) The function f ( x) =
5x − 1
has the line x = 3 as a vertical asymptote.
x −3
x −1
= + ∞ because x − 3 >0 for x>3 and and x − 1 > 0 around x = 3 and
x →3
x →3 x − 3
x −1
lim− f ( x ) = lim−
=−∞
x →3
x →3 x − 3
lim+ f ( x ) = lim+
(e.g.) The function f ( x) =
x( x − 1)
has the lines x = 5 and x = −2 as vertical asymptotes.
( x − 5)( x + 2)
(e.g.) The function f ( x ) =
x −1
has no vertical asymptotes as x 2 + 1 ≠ 0 for any real number x
2
x +1
(e.g.) The function f ( x ) = tan x has infinitely many vertical asymptotes x = ±
As tan x =
2
,±
3π 5π 7π
,±
,±
,
2
2
2
sin x
π 3π 5π 7π
and cos x = 0 for x = ± , ±
,±
,±
,
cos x
2
2
2
2
(e.g.) The function f ( x) =
f ( x) =
π
x2 − 9
has no vertical asymptotes as
x −3
( x − 3)( x + 3)
= x + 3 if
x−3
x ≠ 3 and
lim f ( x ) = lim− f ( x ) = 6
x →3+
x →3
The graph has a “hole” at x =3
NOTE
In order to find the vertical asymptotes (if any) of a rational function of the form
f ( x) =
Polynomial of degree n
find the value(s) of x (if any) that make(s) the denominator equal to
Polynomial of degree m
zero but do not cancels the numerator as well.
Asymptotes pg 1
Last Updated. Feb 15, 2012
2. Horizontal Asymptotes
Definition: A function y = f(x) has the line y =b as a horizontal asymptote if xlim
f ( x ) = b and/or
→+∞
lim f ( x ) = b
x →−∞
Polynomial of degree n
has a horizontal asymptote if
Polynomial of degree m
the degree of denominator is bigger or equal to the degree of numerator m ≥ n
RULE A rational function of the form f ( x) =
•
If m > n then the line y = 0 is horizontal asymptote.
•
If m = n then the equation of the horizontal asymptote is y = (ratio of leading coefficients)
•
f ( x) | = ∞
If m < n then there is no horizontal asymptote as | xlim
→±∞
(e.g) f ( x ) =
5x −1
14
has the line y = 5 (the x-axis) as horizontal asymptote because
= 5+
x−3
x −3
lim f ( x ) = lim
x→ ± ∞
x →±∞
5x −1
=5
x−3
Asymptotes pg 2
Last Updated. Feb 15, 2012
5x2 − 7
5
(e.g.) The function f ( x) = 2
has the horizontal line y = as horizontal asymptote:
9x + 3
9
7
5 x 2 (1 − 2 )
5x2 − 7
5x = 5
lim f ( x ) = lim 2
= lim
x →∞
x →∞ 9 x + 3
x →∞
3
9 x 2 (1 − 2 ) 9
9x
(e.g.) The function f ( x) =
5 x3 − 3
12 x 4 + x 2 + 7
has the horizontal line y = 0 (the x-axis) as horizontal
5 x3 − 3
=0
x →∞ 12 x 4 + x 2 + 7
asymptote: lim f ( x) = lim
x →∞
(e.g.) The function f ( x) =
5 x3 + 3x + 1
12 x 2 + 3
has no horizontal asymptote because:
5 x3 + 3x + 1
5 x3 + 3x + 1
=
+∞
and
lim
f
(
x
)
=
lim
=−∞
x →∞ 12 x 2 + 3
x →−∞
x →− ∞ 12 x 2 + 3
lim f ( x ) = lim
x →∞
For other (than rational) type of function apply the definition directly
(e.g.) The function f ( x) = −2 x + 9 + e− x
lim g ( x) = lim e
x →∞
x →∞
−x
y = −2 x + 9 as the horizontal asymptote because:
= 0 even if lim g ( x) = lim e − x = + ∞
x →−∞
x →−∞
Asymptotes pg 3
Last Updated. Feb 15, 2012
3. Oblique Asymptotes (or Slant Asymptotes)
Definition
A function y = f(x) has the line y = mx+ b as a oblique (or slant) asymptote if f(x) = mx +b + g(x) and
lim g ( x ) = 0
x →±∞
x2 + 1
2
= x +1+
has the line y = x + 1 oblique asymptote as
x −1
x −1
2
2
becomes very small
lim g ( x) = lim
= 0 which means that for x “big enough “the term
x →± ∞
x →± ∞ x − 1
x −1
compared to the term x + 1 and therefore f ( x ) ≈ x + 1.
(e.g.1) The function f ( x) =
← y = x +1
oblique assymptote
← x = 1 vertical assymptote
Note that the above function has x = 1as vertical asymptote:-------
Asymptotes pg 4
Last Updated. Feb 15, 2012
5 x3 + 2 x 2 + 3
3
= 5 x + 2 + 2 has the line y = 5 x + 2 oblique asymptote as
(e.g.2) The function f ( x ) =
2
x
x
1
3
lim g ( x ) = lim(− 2 ) = 0 which means that for x “big enough “the term 2 becomes very small
x →∞
x →∞
x +1
x
compared to the term 5 x + 2 and therefore f ( x ) ≈ 5 x + 2
RULE A rational function of the form f ( x ) =
Polynomial of degree n
has an oblique (slant)
Polynomial of degree m
asymptote iff the degree of numerator is exactly one unit more than the degree of denominator
Degree of numerator = 1+ degree of denominator
NOTE
1. Polynomial functions f ( x ) = an x n + an −1 x n −1 + + a2 x 2 + a1 x + a0 , as well as f ( x) = sin x,
f ( x ) = cos x, have no asymptotes.
2. Exponential functions f ( x ) = e kx has no vertical asymptotes but has y = 0 (x axis) as horizontal
asymptote.
Asymptotes pg 5
Last Updated. Feb 15, 2012
(eg) f ( x ) = e 0.1x has y = 0 (x axis) as horizontal asymptote.
3. Logarithmic functions f ( x) = log a x have only vertical asymptote x = 0 (y-axis):
(e.g.) Graph of f ( x) = ln x
If a >1 then lim+ log a x = − ∞ and if 0< a<1 then lim+ log a x = ∞
x →0
x →0
Asymptotes pg 6
Last Updated. Feb 15, 2012