Horizontal Asymptotes

Horizontal Asymptotes

Horizontal Asymptotes :

Lines of the form y = a, where a is a constant, for which f(x) -> a as f(x)->

or f(x)->-

. X-axis could be a horizontal asymptote.

The following is an example of horizontal asymptote

Consider the function in the example given in the previous page. The graph has been plotted for the function C(t).

To find out the asymptote of the function C(t), we need to find the limit of the function for t tending to .Then, we observe that using the property # 6 of limits

which means that y = 0 .i.e.,x-axis is the horizontal asymptote.

We also need to find the limit of the function for t tending to -. Then, we find that

which means that y = 0 .i.e.,x-axis is the asymptote.

Also, the graph for the function has been redrawn below :

Let us consider another example for horizontal asymptotes

A defense acquisition officer was analyzing the cost data of the purchases his division has made. He found that it is a cumulative distribution function given as :

He wanted to know how the expenditure varied over a given period of time t. So, he wanted to plot the above function. Since it is easy to plot the function if the asymptote is known, he needs to find the asymptotes. To find the asymptote of the function F(t), he needs to find the limit of the function for t tending to

Since the value of e = 2.2718 is greater than 0 and also time period t is greater than 0, the value of e ^-2t becomes 0 when t increases to approximately 4. So, the value of the limit of the function tends to 1 for all values above 4.

The graph for the function F(t) is :