Asymptotes

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
You can find solved problems about asymptotes here

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