Can this be explained in a way a dummy like me can understand, Asymptotes?

Question

Vertical asymptotes are vertical lines near which the function f(x) becomes infinite. If the denominator of a rational function has more factors of (x - a) than the numerator, then the rational function will have a vertical asymptote at x = a.
Horizontal Asymptotes. A horizontal asymptote is a line y = c such that the values of f(x) get increasingly close to the number c as x gets large in either the positive or negative direction. Rational functions have horizontal asymptotes when the degree of the numerator is the same as the degree of the denominator.
Oblique Asymptotes An oblique asymptote is an asymptote of the form y = ax + b with a non-zero. Rational functions have oblique asymptotes if the degree of the numerator is one more than the degree of the denominator. The function g(x) above has an oblique asymptote, namely the line y = x. This is reflected in the last view of the graph of g(x) on a calculator.

READ THE DETAILS, I WANT TO KNOW WHAT THE 3 TYPES OF ASYMTOPES ARE IN NORMAL TERMS.

Answer 1

Look here:
http://id.mind.net/~zona/mmts/functionInstitute/rationalFunctions/oneOverX/oneOverX.html

The green line is the graph of a function, and two red ones are horizontal and vertical asymptotes.
See how the green line gets closer and closer to the red one, but never intersects iit - and it never will. This is was the asymptote is all about - a line, that a function gets closer and closer, but never intersects.
It does not have to be horisontal or vertical - you could rotate the graph on that picture for example - that would be an oblique asymptote...
It doesn't actually have to be a straight line either ... but that's a different topic.

I hope, it helps...

Answer 2

I guess you've transcribed the definition from a textbook, which you're saying you don't understand.

Basically, an asymptote is a line such that, as x approaches a certain value, f(x) approaches a certain line but never actually reaches it. For example, look at the function f(x) = (x + 2) / (x - 3). This has a vertical asymptote at x = 3 because the denominator gets very very small, so the fraction becomes huge. x + 2 is positive on either side of x = 3, so the function races towards positive infinity on both sides of the vertical line x = 3. This function also has a horizontal asymptote at y = 1. This is because, at extremely large and extremely small (negative infinity, not zero) values of x, you can ignore the much smaller + 2 and - 3, and you basically have x / x = 1. Here is a table of values showing f(x) as x increases:

x = 5, f(x) = 3.5
x = 10, f(x) = 1.71
x = 15, f(x) = 1.42
x = 20, f(x) = 1.29
x = 100, f(x) = 1.05
x = 1000, f(x) = 1.005

As you can see, the function initially approaches the asymptote y = 1 pretty quickly. It then slows the approach, because it will never actually reach it.

For an oblique asymptote, consider g(x) = (2x^2 + x + 1) / x. Ignoring + 1 at large values of x gives you 2x + 1 when you divide through by x. The oblique asymptote is 2x + 1. Here's another table, showing the same effect as above, with the values of h(x) = 2x + 1 given as reference.

x = 1, g(x) = 4, h(x) = 2
x = 3, g(x) = 7.33, h(x) = 6
x = 5, g(x) = 11.2, h(x) = 11
x = 10, g(x) = 21.1, h(x) = 21
x = 15, g(x) = 31.07, h(x) = 31
x = 20, g(x) = 41.05, h(x) = 41
x = 100, g(x) = 201.01, h(x) = 201
x = 1000, g(x) = 2001.001, h(x) = 2001

Again, it initally approaches the line 2x + 1 pretty quickly, but then barely gets any closer over a long distance because it can't actually reach that asymptote.

Answer 3

Straight lines that have the property of becoming and staying arbitrarily close to the curve as the distance from the origin increases to infinity. For example, the x-axis is the only asymptote to the graph of sin(x)/x.

A line whose distance to a given curve tends to zero. An asymptote may or may not intersect its associated curve.

If limit x>infinity f(x)=L

then we say that L is a horizontal asymptote of f(x).
--------------------------------------------------------------------------------

An asymptote
A line d is an asymptote of a curve C if and only if the limit of the distance from a point P of C to d is zero, if P recedes indefinitely along the curve.
We say that C has an asymptote d.

Asymptotes and functions.
If the graph of a function has an asymptote d, then we say that the function has an asymptote d.
A function can have more than one asymptote.
If an asymptote is parallel with the y-axis, we call it a vertical asymptote. If an asymptote is parallel with the x-axis, we call it a horizontal asymptote. All other asymptotes are oblique asymptotes.
In this article we only consider functions that are continuous in their domain.
Vertical asymptotes
From the definition we have :

A line x = a is a vertical asymptote of a function f(x)
<=>
(lim f(x) = +infty or -infty ) or (lim f(x) = +infty or -infty )
> a < a


Examples

x+4 x+4
lim ----- = -infty , so x = 3 is a vertical asymptote of -----
< 3 x-3 x-3

x+4 x+4
lim ----- = -infty , so x = 3 is a vertical asymptote of -----
> 3 x-3 x-3

x.x + 3x + 2
lim -------------- = -1 , so x = -2 is not a vertical asymptote.
-2 x + 2


lim tan(x) = +infty , so x = pi/2 is a vertical asymptote of tan(x)
< pi/2

The function tan(x) has many vertical asymptotes.


Horizontal asymptotes
From the definition we have :

A line y = b is a horizontal asymptote of a function f(x)
<=>
lim f(x) = b or lim f(x) = b with b in R
+infty -infty

Examples :

3x2 - 4x -1
lim -------------- = 0.5
infty 6x2 - 6
3x2 - 4x -1
So, y = 0.5 is a horizontal asymptote of a function -------------
6x2 - 6

_______ _______
| 2 | 2
\| x - 1 \| x - 1
lim -------------- = 1 and lim -------------- = -1
+infty x - 1 -infty x - 1

So, y = 1 and y = -1 are horizontal asymptotes.


Oblique asymptotes
Each oblique asymptote d has an equation y = ax + b. Here a and b are unknown real numbers. We'll deduce formulas to calculate a and b from the function f(x).



y = ax + b is oblique asymptote d

<=>

lim |P,Q| = 0
P->infty

<=>

lim |P,Q| = 0
x-> infty
in triangle PQS is |P,Q|=|P,S|.sin(PSQ)
<=>

lim |P,S|.sin(PSQ) = 0
x-> infty

since sin(PSQ) is constant and not zero
<=>

lim |P,S| = 0
x-> infty

since |P,S| = |f(x) - ax - b|
<=>

lim |f(x) - ax - b| = 0
x-> infty

<=>

lim f(x) - ax - b = 0 (*)
x-> infty


From (*) we'll deduce a formula for a

f(x) - ax - b
(*) => lim ----------------- = 0
infty x


<=>

f(x) b
lim ---- - lim a - lim --- = 0
infty x x

<=>


f(x)
(lim ---- ) - a = 0
infty x

<=>


f(x)
lim ---- = a (1)
infty x


With (*) we'll construct a formula for b.

(*) => lim (f(x) - ax ) = b (2)
infty

Since we know a from (1), we can calculate b from (2).
Example :

________
| 2
Take f(x) = \| x - 1 + 2
We calculate a and b.

First, let x -> + infty . Then

________
| 2
\| x - 1 + 2
a = lim ------------------ = ... = 1
x


________
| 2
b = lim \| x - 1 + 2 - 1.x = ... = 2

So the asymptote is y = x + 2 if x -> + infty

Next, let x -> - infty . Then

________
| 2
\| x - 1 + 2
a = lim ------------------ = ... = -1
x

________
| 2
b = lim \| x - 1 + 2 + 1.x = ... = 2

So the asymptote is y = - x + 2 if x -> - infty

Answer 4

Simply put, an asymptote is a value that the function will get very close to, but never reach as you take the limit to infinity.

Think of it as the limiting value of the function. You'll never get past it, you'll only get closer and closer to it.

Without specific functions, that's the best I can explain it.