What Are Vertical Asymptotes

· Mohammad-Ali Bandzar

Vertical asymptotes (often abbreviated to V.A.) are vertical lines that a function will approach (from one or both sides) and become infinitesimally close to, but will never touch. A more formal definition will be included below. For a function to have a vertical asymptote, the range must be infinitely large—that is, it must extend to negative and/or positive infinity.

Three general types of asymptotes exist: vertical asymptotes, horizontal asymptotes, and oblique asymptotes (also commonly called slant asymptotes). In this tutorial we will be focusing on vertical asymptotes.

There are a couple of different ways of visualizing the vertical asymptote. The most common analogy is to visualize it as an airplane approaching a brick wall and how the pilot will pull up/down to avoid a collision. Alternatively, you can just memorize the graph of 1/x and remember that all vertical asymptotes will look roughly the same (the function will either go up or down forever near the asymptote).

Vertical asymptotes exist at the x-value where the denominator of the function is zero and the numerator is non-zero.

1/x graph

Figure 1: Graph of 1/x

Visually we can see that the graph approaches but never reaches x=0 as shown with the dashed line:

1/x graph with v.a

We then want to label our asymptote at the bottom of the page next to the asymptote with its equation:

1/x graph with v.a and label

A Table

Below is a table of our function f(x) approaching our vertical asymptote but never reaching it from the left side. The same would apply to the right as well.

f(x)result
f(-1)-1
f(-0.1)-10
f(-0.01)-100
f(-0.001)-1000
f(-0.0001)-10000
f(-0.00001)-100000
f(-0.000001)-1000000
f(-0.0000001)-10000000

As you can see from the table, our graph approaches infinity as we come close to our asymptote but we are never able to reach/touch/intersect with our asymptote.

If you are ever unsure if there is a vertical asymptote at a specific x-value, I would highly recommend making a table like the one above to verify that the function does in fact approach negative or positive infinity as we approach said value.

Limit Definition

A vertical line x=a (where a is any real number) will be a vertical asymptote of f(x) if any of the following limits are true:

From the left our function approaches infinity:

lim⁡x→a−f(x)=∞\displaystyle\lim_{x \to a^-} f(x) = \infty

Limit from left approaching infinity

From the left our function approaches negative infinity:

lim⁡x→a−f(x)=−∞\displaystyle\lim_{x \to a^-} f(x) = -\infty

Limit from left approaching negative infinity

From the right our function approaches infinity:

lim⁡x→a+f(x)=∞\displaystyle\lim_{x \to a^+} f(x) = \infty

Limit from right approaching infinity

From the right our function approaches negative infinity:

lim⁡x→a+f(x)=−∞\displaystyle\lim_{x \to a^+} f(x) = -\infty

Limit from right approaching negative infinity

Remember that only one (or more) of the above conditions need to be true for it to be a vertical asymptote.

Epsilon Delta Definition

This is a first-year calculus definition which I will not go into depth about here, but the formal proof of a vertical asymptote would be as follows:

∀N>0  ∃δ>0:0<∣x−a∣<δ⇒f(x)>N\forall N > 0 \; \exists \delta > 0 : 0 < |x-a| < \delta \Rightarrow f(x) > N

Or for vertical asymptotes that are approaching negative infinity:

∀N<0  ∃δ>0:0<∣x−a∣<δ⇒f(x)<N\forall N < 0 \; \exists \delta > 0 : 0 < |x-a| < \delta \Rightarrow f(x) < N

If you do not understand the above, that is completely okay. It is basically saying that if I give you any arbitrarily large y value (in this case N) that you can find a segment of the domain so close to our vertical asymptote such that all elements within that interval of the domain would be greater than N.

Pro Tips

  • Remember that the function can never cross/touch a vertical asymptote
  • Make yourself comfortable graphing
  • If you have difficulty graphing, you can always fall back to plotting points and by using the table function on your calculator to generate points quickly
  • Make yourself comfortable solving quadratic and cubic equations
  • Don’t forget to verify that the zero in the denominator is in fact an asymptote
  • Don’t forget that some functions like 1sin⁡(x)\frac{1}{\sin(x)} have an infinite number of vertical asymptotes (it is a periodic function)
  • Use a graphing calculator/desmos.com to verify your solutions
  • Remember that vertical asymptote locations won’t usually be a part of your domain

How to find vertical asymptotes

THANKS FOR READING