The graphs of some functions exhibit unusual behaviour, especially as values of x get larger and larger. Let's have a look at a simple function that has asymptotes.

The strange behaviour of this function and its graph comes from the fact that the variable x is in the denominator.
When we start to fill in values for x, two things are going to happen:
  1. When x is 0, there won't be any value, because division by zero is impossible. This will make the graph discontinuous.
  2. As the value of x gets larger and larger, the value of 1/x gets smaller and smaller, but will never reach zero.
The value of x can of course be any real number, positive or negative. For the moment, let's look only what happens when x is positive and getting larger and larger. Have a look at the table of values on the right:
  1. For x = 0 there is no value; this is shown as ∞
  2. As x gets larger and larger, the fractions 1/x are getting smaller and smaller. They're definitely heading towards zero, but will never get there. The value of 1/x will always be a positive tiny fraction.

    This type of behaviour is called a limit. We can write it as:
      Find out more about limits here.

Here's the partial graph for positive x values only:

You can see the values of y getting smaller and smaller as x increases (direction of arrow); they are getting closer to zero, but will never get there.

This type of behaviour is called asymptotic. The graph seems to be approaching the x axis (in red), which is the line with equation y = 0.

The line y = 0, (in green) is called an asymptote.

Looking again at the graph above you can see that the same behaviour seems to be happening close to x = 0. As x gets closer and closer to zero, from the right, the values of y are getting larger and larger. Since x can't be zero, the value there is also infinity. The y axis, or the vertical line x = 0, is also an asymptote.

Remember that x can be any Real number, and both positive or negative. Let's return to the entire graph of y = 1/x. [You can see this on your Ti calculator:]

You can see that the full graph shows asymptotes in all four directions.

The x and y axes (in green), which are the lines y = 0 and x = 0, become the asymptotes.
Here are the limit statements:




Asymptotes don't have to be the x and y axes:



The asymptotes don't even have to be horizontal or vertical.
Have a look at our page about the hyperbola.

In three dimensions, the asymptotes would be planes; here's an
example of a 3-dimensional graph with one asymptote plane (in green):


This topic is beyond the scope of high school mathematics,
although the base 2D function y = cscθ is in Math 30.


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Content, graphics, HTML & design by Bill Willis 2024