Asymptote, vertical asymptote
An asymptote is a line or curve that comes arbitrarily close to a certain curve. In other words, it is a line close to a given curve, so the distance between the curve and the line tends to zero as the curve reaches higher / lower values. The area of the curve that has an asymptote is asymptotic. Asymptotes are often found in rotation functions, exponentials, and logarithmic functions. An asymptote parallel to the y-axis is called a vertical asymptote.
Determination of the vertical asymptote
If a function f (x) has asymptote (n), the function satisfies the following condition at a finite value C.
In general, a function that is not defined at a finite value has an asymptote. Nevertheless, a function that is undefined at one point cannot have an asymptote at that value if the function is defined in a particular way. Hence it is confirmed by taking the limits at the finite values. If the limits for the finite values (C) approach infinity, the function at C has an asymptote with the equation x = C.
How to find vertical asymptotes - examples
- Consider f ( x ) = 1 / x
The function f ( x ) = 1 / x has both vertical and horizontal asymptotes. f ( x ) is not defined at 0. Therefore, the assumption of the limits at 0 is confirmed.
Note that the function approaching from different directions tends to have different infinities. When approaching from a negative direction, the function tends towards negative infinity; when approaching from a positive direction, it tends to positive infinity. Hence the equation of the asymptote is x = 0.
- Consider the function f ( x ) = 1 / ( x -1) ( x +2)
The function does not exist if x = 1 and x = -2. Therefore, if one takes limits at x = 1 and x = -2, we get
From this we can conclude that the function has vertical asymptotes at x = 1 and x = -2.
- Consider the function f (x) = 3x 2 + e x / (x + 1)
This function has both vertical and oblique asymptotes, but the function does not exist at x = -1. In order to check the existence of an asymptote, the limit values are therefore set at x = -1
Hence the asymptote equation is x = -1.
Another method must be used to find the slanted asymptote.