# Discontinuity: Asymptotic discontinuity

• MHB
• samir
In summary: These are useful in mathematical analysis and calculus, where the notion of infinitesimals can be used to formalize concepts such as derivatives and integrals in a more intuitive way.
samir
Hi again!

I thought I would finish off my previous posts on discontinuity by discussing asymptotic discontinuity. So let's focus on these alone in this thread.

I'm not familiar with the origin of the term "asymptote", but from what I can tell, it has asymptotic discontinuity to thank for for its existence. That is to say asymptotic discontinuity is the reason we have these vertical and horizontal lines that we call asymptotes.

So what is asymptotic discontinuity? The simplest explanation I can think of is that it's the point at which a function is discontinuous because the limit at the given point is unreachable, because it's infinite!

Assume we have the following function.

$$f(x)=\frac{1}{x}$$

How do we show that this function has vertical and horizontal asymptotes? My current understanding is that we have to show that the limit is infinite. But how do we do that?

For the vertical asymptote, I believe that we can write something like this:

$$\lim_{{x}\to{0^{\pm}}}f(x) \rightarrow \frac{1}{x}= \infty$$

Letting x approach 0 implies that the quotient approaches infinity.

But I don't think we can write this:

$$\lim_{{x}\to{0^{\pm}}}f(x)= \infty$$

Because x is not allowed to be equal to zero. The function is undefined at x=0.

$$\lim_{{x}\to{0^{\pm}}}f(x)= \text{undefined}$$

Or am I looking at this the wrong way? Is the limit in fact equal to infinity? Surely nothing can really be equal to infinity as infinity is not even a number?

What about the horizontal asymptote? How do we show that this function has a vertical asymptote? Do we have to solve for the dependent variable and kind of do the same thing in reverse?

Now, is it only rational functions that have asymptotes? I have mostly seen this type of function having an asymptote. I know tangent function also has asymptotes. But that has to do with the definition of what a tangent function is. We can't divide by zero.

Can a function have a vertical asymptote but no horizontal asymptote? I would say yes.

Can a function have a vertical asymptote and a horizontal asymptote? I would say yes to that too.

But can a function have a horizontal asymptote alone, and no vertical asymptote? I'm not so sure of that.

Last edited:
samir said:
The simplest explanation I can think of is that it's the point at which a function is discontinuous because the limit at the given point is unreachable, because it's infinite!

That's true for vertical asymptotes and they occur as the function approaches some constant value. For horizontal asymptotes the limit is some constant, for example

$$\lim_{x\to\infty}\tan^{-1}(x)=\dfrac{\pi}{2}$$

samir said:
Can a function have a vertical asymptote but no horizontal asymptote?

Yes. $\tan(x)$ is an example.

samir said:
Can a function have a vertical asymptote and a horizontal asymptote? I would say yes to that too.

That's correct.

samir said:
But can a function have a horizontal asymptote alone, and no vertical asymptote? I'm not so sure of that.

Yes, it can. $\tan^{-1}(x)$ is such a function.

***

$\lim_{x\to0^\pm}\dfrac1x=\pm\infty$ and it's purely a notational convention. As you say, infinity is not a number and can't be treated as such in the sense of real numbers. There are systems, such as the extended reals, where infinities are treated somewhat differently.

A bit of humor:

"Infinity isn't a number, it's a place."

-"What place is that?"

"Not anywhere near here."

The notion $\lim\limits_{x \to 0^+} \dfrac{1}{x} = \infty$ is actually well-defined, and can be stated without mentioning infinity:

"For any positive (real) number $M$ there exists a positive real number $\delta$ such that if $0 < x < \delta$, then $\dfrac{1}{x} > M$".

Informally, "$\dfrac{1}{x}$ grows without bound as $x$ nears 0 from the right."

greg1313 said:
There are systems, such as the extended reals, where infinities are treated somewhat differently.

I especially like the sentence:
With these definitions $$\displaystyle \overline{\mathbb{R}}$$ is not a field, nor a ring, and not even a group or semigroup. However, it still has several convenient properties.

In other words, it violates all known algebraic structures, but it's still useful. (Wink)

I like Serena said:
I especially like the sentence:
With these definitions $$\displaystyle \overline{\mathbb{R}}$$ is not a field, nor a ring, and not even a group or semigroup. However, it still has several convenient properties.

In other words, it violates all known algebraic structures, but it's still useful. (Wink)

One can also form the *hyperreals*, which not only contain "infinities", but also *still* remain a field.

## What is an asymptotic discontinuity?

An asymptotic discontinuity refers to a point at which a mathematical function or curve approaches infinity or a finite limit without actually reaching it. This results in a sudden jump or break in the function, creating a discontinuity.

## How is an asymptotic discontinuity different from other types of discontinuities?

An asymptotic discontinuity is different from other types of discontinuities, such as jump or removable discontinuities, because it does not result in a sudden change in the value of the function. Instead, it represents a point at which the function becomes undefined or infinite.

## What causes an asymptotic discontinuity?

An asymptotic discontinuity is caused by a function or curve approaching a vertical asymptote or a point of infinite slope. This can occur when the function has a denominator that approaches zero or when there is a sharp change in the behavior of the function.

## What are some real-world examples of asymptotic discontinuities?

Asymptotic discontinuities can be observed in various real-world scenarios, such as in the behavior of a population growth curve, the temperature distribution along a metal rod, or the rate of change of a chemical reaction over time.

## How are asymptotic discontinuities used in mathematics and science?

Asymptotic discontinuities are commonly used in mathematics and science to model and analyze complex systems, predict trends and patterns, and identify critical points in a function. They are also used in calculus to evaluate integrals and determine the convergence of infinite series.

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