How to Find Asymptotes of a Curve?

For any curve represented as b = h (a) or h (a, b) = 0, we can define Asymptote as a straight line (can be horizontal, vertical or oblique) whose distance from the curve tends to 0 when coordinates lying on the curve tend to infinity (positive or negative). There can be 3 types of asymptotes possible for curves: vertical asymptote, horizontal asymptote and oblique asymptote. Algebraically it is possible to find out the asymptotes to any curve (especially horizontal and oblique). So, let us see how to find asymptotes of a curve.

1st substitute 'b' equal to ma + c in the equation of curve and assemble the result in form:

D_n aⁿ + D_{n – 1} a^{n – 1} + D_{n – 2} a^{n – 2} +……. + D₁ a + D₀ = 0................ Equation --- (1)

Next we solve the equation simultaneously:

D_n = 0 & D_{n – 1} = 0,

For every couple of answers of m and c, frame the equation of an asymptote as: b = ma + c.

If there is no D_{n – 1} term in 1st equation, then solve:

           D_n = 0 & D_{n – 2} = 0,

The equation D_n aⁿ + D_{n – 1} a^{n – 1} + D_{n – 2} a^{n – 2} +……. + D₁ a + D⁰ = 0 has two roots at infinity.

Next substitute b = 1/a. We get:

D₀bⁿ + D₁ b^{n – 1} + D₂ b^{n – 2} +……. + D_{n – 1} b + D_n = 0,

Here also the equation has two roots at b = 0. This means, the equation is of the form:

D₀ bⁿ + D₁ b^{n – 1} +……. + D_{n – 2} b² = 0

b² (D₀ b^{n – 2} + D₁ b^{n – 3} +……. + D_{n – 2}) = 0

Therefore, we have:

           D_n = 0 & D_{n – 2} = 0