What is a Horizontal Asymptote
An asymptote is a line or curve that become arbitrarily close to a given curve. In another words it is a line close to a given curve, such that the distance between the curve and the line approaches zero when the curve is reaching higher/ lower values. The region of the curve that has an asymptote is asymptotic. Asymptotes are often found in rotational functions, exponential function and logarithmic functions. Asymptote parallel to the xaxis is known as a horizontal axis.
How to Find the Horizontal Asymptote
An asymptote exists if the function of a curve is satisfying following condition. If f(x) is the curve, then a horizontal asymptote exist if,
Then horizontal asymptotes exist with equationy=C. If the function approaches finite value (C)at infinity, the function has an asymptote at that valueand the equation of an asymptote is y=C. A curve may intersect this line at several points, but 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 form f(x)=a^{x} and [a>0]
Exponential functions are the simplest examples of horizontal asymptotes.
Taking the limits of the function at positive and negative infinities gives, lim_{x→∞ }a^{x} =+∞ and lim_{x→∞ }a^{x} =0. The right limit is not a finite number and tends to positive infinity, but the left limit approaches the finite values 0.
Therefore, we can say that exponential function f(x)=a^{x} has a horizontal asymptote at 0. The equation of the asymptote line is y=0, which also is the xaxis. Since a is any positive number, we can consider this as a general result.
When a=e= 2.718281828, the function is also known as the exponential function. f(x)=e^{x} has specific characteristics and 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 known as a rational function. Rational function may have both vertical and horizontal asymptotes.
i. Consider the function f(x)=1/x
Function f(x)=1/x has both vertical and horizontal asymptotes.
To find the horizontal asymptote find the limits at infinity.
lim_{x→}=+∞ 1/x=0^{+ }and lim_{x→}=∞ 1/x =0^{–}
When x→+∞, function approaches 0 from the positive side and when x→=∞ function approaches 0 from the negative direction.
Since function has a finite value 0 when approaching infinities, we can deduce that the asymptote is y=0.
ii. Consider the function f(x)=4x/(x^{2}+1)
Again find the limits at infinity to determine the horizontal asymptote.
Again the function has asymptote y=0, also in this case the function intersects the asymptote line at x=0
iii. Consider the function f(x)=(5x^{2}+1)/(x^{2}+1)
Taking the limits at infinity gives,
Therefore, the function has finite limits at 5. So, the asymptote is y=5
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