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Calculus I









© The scientific sentence. 2010

Calculus I: Asymptotes





Asymptotes are useful guides to complete the graph of a function. An asymptote is a line to which the curve of the function approaches at infinity or at certain points of discontinuity.

There are three types of asymptotes: vertical asymptotes, horizontal asymptotes and oblique asymptotes.

1. Vertical asymptote

A line x = a is a vertical asymptote of the graph of the function f if either:

lim f(x) = ± ∞     or     lim f(x) = ± ∞
x → a+     x → a-

Note:

The vertical asymbtotes exist only for the points of discontinuity of the function.

For a rational irreducible function, every value of x that make zero the denominator is a verticale asymptote for this function
.

A polynomial function doesn't have a verical asymptote. A function can have one or more vertical asymptotes


Example

f(x) = (x + 1)/(x - 1)(x + 2)

This fonction is rational. The points x = 1 and x = - 2 are the points of discontinuity.

At x = 1, we have:
lim f(x) = 2/0
x → 1
= 2/(3)0+ = + ∞
x → 1+
= 2/(3)0- = - ∞
x → 1-

Therefore

x = 1 is a vertical asymptote.

At x = - 2, we have:
lim f(x) = - 1/0
x → - 2
= - 1/(- 3)0+ = + ∞
x → - 2+
= - 1/(- 3)0- = - ∞
x → - 2-

Therefore

x = - 2 is a vertical asymptote.



2. Horizontal asymptote

A line y = b is a horizontal asymptote of the graph of the function f if either:

lim f(x) = b or lim f(x) = b
x → + ∞ x → - ∞

Example

f(x) = (x + 1)/(x - 1)

We have
lim f(x) = lim (x + 1)/(x - 1) = 1
x → + ∞

And
lim f(x) = lim (x + 1)/(x - 1) = 1
x → - ∞

Therefore
y = + 1 is a horizontal asymptote.

To sketch the graph near this asymptote, we also determine the left and right limit around the value y = 1. Then:

lim f(x) = lim (1 + 1/x)/(1 - 1/x) = (1 + 0+)/(1 - 0+) =
1+/1- = 1+
x → + ∞
And
lim f(x) = lim (1 + 1/x)/(1 - 1/x) = (1 + 0-)/(1 - 0-) =
1-/1+ = 1-
x → - ∞



Note:

A polynomial function doesn't have a horizontal asymptote.

A rational function can have a horizontal asymptote if the degree of the numerator is less than the degree of the denominator.

A function can have 0, 1, or 2 horizontal asymptotes. never more than 2.


3. Oblique asymptote

A graph of a function f has an oblique asymptote y = a x + b if
a) f(x) can take the expression: f(x) = a x + b + g(x), and
b) lim g(x) = 0     or     lim g(x) = 0
x → + ∞     x → - + ∞

Example



f(x) = (2 x 2 + x - 2)(x - 1)

f(x) takes the form:

f(x) = 2 x + 3 + g(x),
where
g(x) = 1/(x - 1)

We have

lim g(x) = 0+lim f(x) = + ∞
x → + ∞x → + ∞

lim g(x) = 0-lim f(x) = - ∞
x → - ∞x → - ∞

lim g(x) = 1/0+ = + ∞ → lim f(x) = + ∞
x → 1+ x → 1+

lim g(x) = 1/0- = - ∞ → lim f(x) = - ∞
x → 1- x → 1-


f(x) = 0 for x1 = (- 1 + √17)/4, and x2 = (- 1 - √17)/4

The ligne of equation y = 2 x + 3 in an oblique asymptote.


Remark:

At near infinity (±∞), the function f(x) behaves as the liear function y = 2 x + 3, and
At near x = - 1, the fuction f(x) behaves as the function g)x) = 1/(x - 1).

We say the dominamt term is 2 x + 3 when x is large and g(x) at the neighborhood of x = 1.


Note:

A polynomial function doesn't have an oblique asymptote.

Any function cannot have more than two oblique asymptotes.

A rational function has an oblique asymptote if the degree of the numerator is equal to the degree of the denominator plus one.


A rational function cannot have a horizontal asymptote and an oblique asymptote at the same time.



4. Exercices





  


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