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How To Find Horizontal Asymptotes?

A horizontal asymptote defines how a function works at the edges of a graph. It is a horizontal line, and the function can also cross the asymptote and touch it. A horizontal asymptote is defined for functions where the numerator and the denominator are polynomials. Before learning to find the horizontal asymptote, it is essential to know what a function is. It is an equation that describes how things are related. Visually it can be represented on a graph and tells the relation between X and Y.

In algebraic terms, the horizontal asymptote for a function is a line which is horizontal and is a function on the graph as the x co-ordinates approach ∞ or -∞. If y = k is a horizontal asymptote for y=f(x), then y-coordinates of ‘f(x)’ gets nearer to k as the curve moves to the right (x→ ∞) or the left (x → -∞).

What is an Asymptote?
It is a line wherein the function gets close to but never touches, but symbolically, x it can touch at infinity. x= infinity.

A function can have no asymptotes or have one or two. If you are given a graph and have to find the asymptotes, then look at both the left and right side of the graph. If the curve levels off, then find the ‘y-coordinate’ to which the curve is leading to, and that enables to trace the horizontal line at the required height the asymptote should be located.

Horizontal Asymptote can be Found Analytically:
If the functions are rational and have the form f(x) = p(x) / q(x) where both the numerator and the denominator are polynomials then highest order term analysis and p(x) have the greatest degree. To perform the highest order term analysis on a function with rational numbers, first expand both the polynomials (numerator and denominator) and write a function that has only terms of highest order while ignoring the lower order terms. Common variables and factors can also be overlooked or cancelled. For a constant k, y=k is the horizontal asymptote when both the top and bottom degree matches. If the result has powers of x left on top, then no horizontal asymptote is present. If the result has the power of ‘x’ at the bottom, then there is one horizontal asymptote as y = 0.

Trigonometric Functions:

Finding Asymptote when dealing with trigonometric functions is simple. You can following the same steps to find horizontal asymptote of rational functions can be done.  But the trig functions are cyclical so we need to deal with more asymptotes.

Real-life Applications of Horizontal Asymptotes:
Exponential cooling is one of the real-life examples of negative horizontal asymptote. If a piping hot object is kept in a room colder than the hot object, the temperature of the object will go down exponentially and reduce to the room temperature. If the temperature of the room is in negative, then it is a negative asymptote. This theory works if the room is large and the object cannot heat the room.

Horizontal asymptotes are not difficult, in fact, they are easy to find if you learn how to understand it through a graph. In case the graph is not available you can use the highest order term analysis for rational function or even the exponential functions.