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.
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