Asymptotes

An Asymptote is a straight line that constantly approaches a given curve but does not meet at any infinite distance. In other words, Asymptote is a line that a curve approaches as it moves towards infinity.AsymptotesHow to find asymptotes

The curves visits these asymptotes but never overtake them. The method opted to find the horizontal asymptote changes based on how the degrees of the polynomials in the numerator and denominator of the function are compared. If both the polynomials have the same degree, divide the coefficients of the largest degree terms.

What is an asymptote of a curve?

An asymptote of the curve y = f(x) or in implicit form: f(x,y) = 0 is a straight line such that the distance between the curve and the straight line lends to zero when the points on the curve approach infinity.

There are three types of asymptotes namely:

  • Vertical Asymptotes
  • Horizontal Asymptotes
  • Oblique Asymptotes

The point to note is that the distance between the curve and the asymptote tends to be zero as it moves to infinity or -infinity.

Horizontal AsymptoteAsymptotesWhen x moves to infinity or -infinity, the curve approaches some constant value b, and called as Horizontal Asymptote.

Vertical AsymptoteAsymptotesWhen x approaches some constant value c from left or right, the curve moves towards infinity, or -infinity and this is called as Vertical Asymptote.

Oblique AsymptoteAsymptotesWhen x moves towards infinity or -infinity, the curve moves towards a line y = mx + b, called as Oblique Asymptote.

Please note that m is not zero since that is a Horizontal Asymptote.

How to find horizontal asymptotes

To recall that an asymptote is a line that the graph of a function visits but never touches. In the following example, a Rational function consists asymptotes.AsymptotesIn the above example, we have a vertical asymptote at x = 3 and a horizontal asymptote at y = 1. The curves approaches these asymptotes but never visits them.

The method to identify the horizontal asymptote changes based on how the degrees of the polynomial in the function’s numerator and denominator are compared.

Example :

Since both the polynomials are second degree, therefore the asymptote lies at y = 6/2 or y = 3.

When the polynomial in the numerator has a lower degree than the denominator, the x-axis at y = 0 is the horizontal asymptote.

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Practise This Question

The sum of first n consecutive odd numbers is equal to n2.