## What is a horizontal asymptote?

An asymptote is a line or curve that comes arbitrarily close to a certain curve. In other words, it is a line close to a given curve, so the distance between the curve and the line tends to zero as the curve reaches higher / lower values. The area of the curve that has an asymptote is asymptotic. Asymptotes are often found in rotation functions, exponentials, and logarithmic functions. An asymptote parallel to the x-axis is referred to as the horizontal axis.

## How to find the horizontal asymptote

An asymptote is present when the function of a curve fulfills the following condition. If f (x) is the curve, then there is a horizontal asymptote if,

Then there exist horizontal asymptotes with equationy = C. If the function approaches a finite value (C) at infinity, the function has an asymptote at that value and the equation of an asymptote is y = C. A curve can intersect this line at multiple points, but it becomes asymptotic as it approaches infinity.

To find the asymptote of a given function, find the limits at infinity.

## Finding Horizontal Asymptotes - Examples

### Exponential functions of the form f (x) = a

^{x}and [a> 0]

Exponential functions are the simplest examples of horizontal asymptotes.

If you take the limits of the function in positive and negative infinite, you get lim _{x → -∞} a ^{x} = + ∞ and lim _{x → -∞} a ^{x} = 0. The right limit value is not a finite number and tends towards positive infinity, but the left limit value approaches the finite values 0.

Hence we can say that the exponential function f (x) = a ^{x has} a horizontal asymptote at 0. The equation of the asymptote line is y = 0, which is also the x-axis. Since a is any positive number, we can take this as a general result.

If a = e = 2.718281828, the function is also referred to as the exponential function. f (x) = e ^{x} has specific properties and is therefore important in mathematics.

### Rational functions

A function of the form f (x) = h (x) / g (x) where h (x), g (x) are polynomials and g (x) ≠ 0 is called a rational function. The rational function can have both vertical and horizontal asymptotes.

I. Consider the function f (x) = 1 / x

The function f (x) = 1 / x has both vertical and horizontal asymptotes. To find the horizontal asymptote, find the boundaries at infinity. lim _{x →} = + ∞ 1 / x = 0 ^{+} and lim _{x →} = -∞ 1 / x = 0 ^{-} If x → + ∞, function against 0 from the positive side and if x → = -∞ function against 0 from the negative direction. Since the function has the finite value 0 as it approaches infinity, we can deduce that the asymptote is y = 0.

ii. Consider the function f (x) = 4x / (x ^{2} +1)

Again, find the boundaries at infinity to determine the horizontal asymptote.

Again the function has the asymptote y = 0, in this case too the function intersects the asymptote at x = 0

iii. Consider the function f (x) = (5x ^{2} +1) / (x ^{2} +1)

To take the boundaries in infinity gives

Therefore the function has finite limits at 5. The asymptote is therefore y = 5