How To Find Vertical Asymptotes?

An asymptote is a line that gets close but never reaches it. Asymptotes are of three types namely horizontal, vertical and oblique. The vertical asymptote is the easiest and common to ascertain. Here you will learn about vertical asymptotes. It is abbreviated as VA for a function is a line which is vertical and if k is a constant then x= k where f(x) which is a function is unbounded. So, y becomes large, and y tends to infinity (y→ ∞) or in a negative direction (y→ -∞) as x reaches k from left or right.

You can imagine vertical asymptote to be a like a brick wall that you cannot cross or think of it as a huge mountain. If you cannot move left or right from the mountain, the only way to traverse it is by flying over it. If you imagine that the mountain is too high vertically, then you will end up flying forever and yet not get over it!

A vertical asymptote is equal to a line that has an infinite slope. It is a rational function which is found at the x coordinate, and that makes the denominator of the function to 0. A rational function is a polynomial equation.

Finding vertical asymptotes:
The VA is the easiest and the most common, and there are certain conditions to calculate if a function is a vertical asymptote.
It can be calculated in two ways:

• Graph: If the graph is given the VA can be found using it. If it looks like a function that is towards the vertical, then it can be a VA. To check if it is a VA or not, you can draw a vertical line at the x-axis, and if the line touches any part of the graph then it is an asymptote, and if it doesn’t, it is a VA.
• From equations: There can be two different functions, and these are trigonometric functions and rational functions.
• Rational functions: If the function you have is a rational function of ‘f(x)’ then
  f(x) = p(x) / q(x) where p and q are polynomials then to find the VA.
  Numerator and denominator should be factored and reduced. Once done, check the denominator factors. If there is a factor (x-a), then it is a VA and if there is a factor (x+a) then also there is a VA.
• For trigonometric functions: This factoring method is used only on rational functions. Tan x, cot x, sec x, and cosec x have VA and that too infinite number while the other two standard trigonometric functions don’t have it.

Asymptotes are properties of lines that depict the way the function behaves. The VA represents that the function has y values that are unbounded. It is essential to find them either through a given graph or through a function analytically using the equation of a function. Also, it needs to be noted that the graph is a physical representation of a mathematical entity. A geometric line has no width, and thus a line can get closer without every coinciding.

Asymptotes are lines that explain the behavior of functions.  A student may think that calculations based on asymptote are tough, but with proper practice, these techniques can be understood and comprehended easily.  All you need to do is understand and apply it in the right way.