What are the Asymptotes?

Overview

A line that a curve approaches is known as asymptote. Any graph (curve) approaches to it but never touches it.

There are total three types of asymptotes-

1.     Vertical asymptote

2.     Horizontal asymptote

3.     Oblique/slant asymptote

Asymptotes

An asymptote of a curve of function y = f(x) is a line which does not intersect on the graph.

Types of asymptotes-

1. Vertical asymptote-

A line x = a which is straight is a vertical asymptote of the graph of the function y = f(x) if atleast one of the following condition does it follow-

2. Horizontal asymptote-

A line y = L is called horizontal asymptote to the curve of the function f(x)-

If f(x)→ L as x→∞ or as x→-∞

Example: Find the horizontal asymptote of the function-

Sol. In order to find the horizontal asymptote-

Hence the horizontal asymptote is y = 2.

3. Oblique asymptote or slant asymptote-

A straight line y = mx + c where m ≠ 0 will be an oblique/slant asymptote to the graph of the function ‘f’ if-

Note- the value of m can be find as-

And the value of c can be find as-

Example: Find the slant asymptote of the function f(x) = x + x .

Sol. Find the value of m-

Hence the y = x + c,

Now find c

Here c must not be infinite.

So we can say that f does not have a slant asymptote at ∞.

Procedure to find the asymptotes parallel to axes of a polynomial function-

Theorem-

Suppose f(x , y) is a polynomial in x and y.

A straight line y = c is an asymptote of a curve f(x , y) = 0 if an only if y – c is a factor of the co-efficient of the highest power of x in f(x , y).

Example: Determine the asymptotes parallel to axes of the curve:

Sol. The given function can be written as-

Asymptote parallel to x-axis-

Equating the coefficient of the highest power of x to zero, we get y² = 0 which means y = 0. This is an asymptote.

Asymptote parallel to y-axis-

Equating the coefficient of the highest power of x to zero, we get x² – a² = 0

Which gives x = ±a , that means x = +a , x = -a     

So that these are the asymptotes.

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