# Asymptote

A asymptote It is a straight line that continuously approaches the curve of a function, but never touches it as the curve advances infinity in one direction, that is, the distance between the two tends to be zero, as they extend indefinitely.

Figure I shows the graphical representation of the function f (x) = 1 / x; the axes "x" and "y" are the asymptotes of the function. As x approaches infinity, the curve gets closer and closer to the x axis, but never touches it. Similarly, as the y-axis approaches infinity, the curve also gets closer and closer to the axis, but never touches it.

## Procedure to find asymptotes:

### Procedure 1:

1. Substitute mx + c = y into the equation of the curve and match the coefficients of the two highest powers of x to zero.
2. Determine the values of myca from the equations. If m1, c1, m2, c2, Y1= m1 x + c1 ; Y2 = m2 x + c2; … Yn = mn x + cn

Example: Find the asymptotes of the x-curves3 + 2x2 - (xy)2  - (2 and)3 + xy - y2 + 1 = 0

Substituting mx + c = y in the equation we obtain

x3 + 2x2 - [x (mx + c)]2 - [2 (mx + c)]3 + x (mx + c) - [(mx + c)]2 + 1 = 0

Equalizing the coefficients x to zero3 (higher grade), we have:

1 + 2m - m2 - 2m3 = 0 → (1 + 2m) (1 - m2) = 0

Thus:

m = 1; -1 ; -1/2

Equalizing the coefficients x to zero2, we have:

2c [1 - m - (3m)2] + m - m2 = 0

Thus:

m = 1 and c = 0; when m = -1 → c = -1; and when m = -1/2 → c = 1/2

The asymptotes will be:

y = x; y = - (x + 1); y = ½ - (x / 2)

### Procedure 2 (Short method):

If we substitute x = 1 and y = m in the highest degree terms of the equation we obtain ϕn(m); ϕn-1(m) is obtained by substituting x = 1 and y = m in the (n-1) term. Equaling ϕn(m) = 0, we obtain the roots as m1, m2,… Mn. Values corresponding to c1, c2,… Cn is obtained by substituting using the formula:

cn= [ϕn-1(m)] / [ϕn(m)]

Therefore the asymptotes will be and1 = m1x + c1; Y2 = m2x + c2… Yn = mnx + cn

Example: Find the asymptotes of the x-curves3 + 2x2 - (xy)2 - (2 and)3 + xy - y2 + 1 = 0

Substituting x = 1 and y = m in the equation we obtain:

ϕ3(m) = 1 + 2m - m2 - 2m3      ; ϕ2(m) = m - m2