An asymptote is a line that runs very close to a curve but the curve never meets the line. At infinity the curve coincides with the asymptote. In other words, an asymptote of a curve is a line such that the distance between the curve and the line tends to zero. The asymptote gets very very close to the curve but never touches it.
Vertical asymptote:
If the above defined asymptote is vertical, that means, that if the asymptote is parallel to the y axis it is called the vertical asymptote. The general equation of the vertical asymptote is y = a, where a is any real number.
How to find vertical asymptotes:
The simplest way to find vertical asymptotes of functions is to graph them. Consider for example that we need to find the vertical asymptote of the function: f(x) = 1/x. Let us graph this function and see what we get. To be able to graph the function we first need the table of values. So let us make a table of value of x and the corresponding values of f(x).
Now let us plot those values on a graph.
Vertical asymptotes can also be found algebraically. In case of rational functions, there is a denominator. If we equate the denominator to zero and then solve for x, the values of x that we thus get would be the vertical asymptote. Let us try to understand that with an example:
Find the vertical asymptotes of the function f(x) = (2x^2-4x+5)/(x^2-2x+1)
Solution: In such a function, we shall work only on the denominator. Here the denominator is x^2 - 2x + 1. Therefore we equate that to zero. So we have:
x^2 - 2x + 1 = 0 now we factor the left hand side
(x-1)(x-1) = 0 now use the zero product rule
(x-1) = 0, (x-1) = 0 now solve for x
X = 1. That is the equation of the vertical asymptote
