How to Find Horizontal Asymptotes?

Some of the students face difficulties to find asymptotes when they get introduced to them for the first time. It takes time to get habituated with this. If the formula is explained complicatedly, then it becomes much tougher for them to calculate or memorize. Here is an easy method that can be easily digested and can stay with you for a long time.

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First, let me give you a very succinct description of asymptotes. There are two types of Asymptotes – one is virtual and the other is horizontal. Both approach in different ways. Virtual asymptotes indicate a specific behaviour but horizontal asymptotes indicate a general behaviour. Another important difference is that you cannot touch virtual asymptotes but you touch and even cross horizontal asymptotes. To know how to find horizontal asymptotes on the graph, an example will be included for you.

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Horizontal Asymptotes are simply horizontal lines on the graph. You can find horizontal asymptotes in the following ways –

1. If the degree of the numerator is less than the denominator, then the horizontal asymptote will be on the x-axis (y = 0).
2. If the degree of the numerator is greater than the denominator, then the horizontal asymptote will be “none” which means there is no horizontal line.
3. There is another situation where the denominator and numerator are the same. The horizontal asymptote, y = lead number of the numerator/lead number of the denominator.

To simplify this, the horizontal asymptote will be like this.

If m < n, y =0 where m = numerator, n = denominator

If m > n, none where m = numerator, n = denominator

If m = n, y = a/b where m = numerator, n = denominator

Examples  

Where m < n

1. f (x) = 4x + 7 / 6x² – 5

Here the degree of the numerator is less than the denominator, m < n so the horizontal asymptote will be y = 0

2. f (x) = x / x⁴ – 1

Here the degree of the numerator is less than the denominator, m < n so the horizontal asymptote will be y = 0

Where m > n

1. f (x) = 5x² / 3 + x

Here the degree of the numerator is greater than the degree of the denominator, m > n so the horizontal asymptote will be “None”.

2. f (x) = 5x³ / x² – 4x + 2

Here the degree of the numerator is greater than the degree of the denominator, m > n so the horizontal asymptote will be “None”.

Where m = n

1. f (x) = 3x² + 2x + 2 / 2x² + 6x + 7

Here the numerator is equal to the denominator, m = n so the horizontal asymptote will be y = a/b

Y = 3/2

2. f (x) = 3x² + x / x² + 4

here the numerator is equal to the denominato, m = n so the horizontal asymptote will be y = a/b

Y = 3