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:**

- Substitute mx + c = y into the equation of the curve and match the coefficients of the two highest powers of x to zero.
- Determine the values of myca from the equations. If m
_{1}, c_{1}, m_{2}, c_{2}, Y_{1}= m_{1}x + c_{1}; Y_{2}= m_{2 }x + c_{2}; … Y_{n}= m_{n}x + c_{n}

**Example:** Find the asymptotes of the x-curves^{3 }+ 2x^{2 }- (xy)^{2 } - (2 and)^{3 }+ xy - y^{2} + 1 = 0

Substituting mx + c = y in the equation we obtain

x^{3} + 2x^{2} - [x (mx + c)]^{2} - [2 (mx + c)]^{3 }+ x (mx + c) - [(mx + c)]^{2 }+ 1 = 0

Equalizing the coefficients x to zero^{3} (higher grade), we have:

1 + 2m - m^{2} - 2m^{3} = 0 → (1 + 2m) (1 - m^{2}) = 0

Thus:

m = 1; -1 ; -1/2

Equalizing the coefficients x to zero^{2}, we have:

2c [1 - m - (3m)^{2}] + m - m^{2} = 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 ϕ_{n}(m); ϕ_{n-1}(m) is obtained by substituting x = 1 and y = m in the (n-1) term. Equaling ϕ_{n}(m) = 0, we obtain the roots as m_{1}, m_{2},… M_{n}. Values corresponding to c_{1}, c_{2},… C_{n} is obtained by substituting using the formula:

c_{n}= [ϕ_{n-1}(m)] / [ϕ_{n}(m)]

Therefore the asymptotes will be and_{1 }= m_{1}x + c_{1}; Y_{2} = m_{2}x + c_{2}… Y_{n} = m_{n}x + c_{n}

**Example:** Find the asymptotes of the x-curves^{3 }+ 2x^{2} - (xy)^{2} - (2 and)^{3} + xy - y^{2} + 1 = 0

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

ϕ_{3}(m) = 1 + 2m - m^{2} - 2m^{3} ; ϕ_{2}(m) = m - m^{2}