Limits at Infinity and Asymptotes

Infinity is an intriguing concept to explore, especially looking at what happens to functions as values approach infinity, i.e., limits at infinity. While exploring limits at infinity, you will encounter another mathematical concept: asymptotes. No, 'asymptotes' are not another type of tote bag, like the one you find at your school or university's gift shop. This strange-sounding term 'asymptote', at least in mathematics, is a straight line that accompanies certain functions and has some intriguing properties. Let's take a look.

Limits at Infinity and Asymptotes meaning

So, what happens to different functions as they approach infinity? Well, as a function approaches infinity, it gets closer to its limit at infinity.

If a function \(y=f(x)\) is given, the limit of this function at infinity is not simply \(\infty\) itself. The word 'limit' is crucial here. When finding the limit of a function, you are not calculating the value of the function at infinity, which is mathematical nonsense, as infinity is not a number.

It is all about knowing what happens to the function \(f(x)\) as the value of \(x\) approaches infinity. There are two potential outcomes as the input to a function approaches infinity: either the limit diverges to infinity or converges to a certain value.

Figure 1.- Limit of a function approaching a certain value at infinity

From the above diagram, you can see that as \(x \rightarrow \infty\), the function \(f(x)\) reaches a constant, which is bounded by the line \(y=l\). We write it as:

$$\lim_{x \to \infty} f(x)= l$$

Hence, the limit of the function is \(l\).

In other words, the function converges to \(l\) but never reaches it; it only gets closer and closer.

Some typical examples are the exponential and the logarithmic functions. The straight line that bounds these values is known as an asymptote. An asymptote is rigorously defined as follows:

In simpler terms, an asymptote is an imaginary line that a curve approaches as its input tends to infinite. Remember that the asymptote and the curve never actually meet, but at infinite, they are infinitely close.

There are three types of asymptotes:

Connection between limits at infinity and Horizontal Asymptotes

As the name suggests, horizontal asymptotes are horizontal, i.e. parallel to the \(x\)-axis. The slope of any horizontal line is \(0\). Let's take a look at the example of a decreasing function again:

Figure 2.- The Horizontal Asymptote to a curve defined by \(y=f(x)\)

\[\lim_{x \to +\infty} f(x)=b \]

• If the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator, then no horizontal asymptote exists for that curve.

• If the degree of the polynomial in the numerator is lower than the degree of the polynomial in the denominator, then the equation of the horizontal is always \(y=0\).

• If the degree of the numerator is equal to the degree of the denominator, then divide by the leading coefficient (the number multiplied by the variable of the highest degree) of the numerator by the leading coefficient of the denominator. The quotient of these leading coefficients is the y value of the horizontal asymptote.

Vertical asymptotes and limit at infinity calculation

Vertical asymptotes are asymptotes which are parallel to the y-axis. In this case, for a finite value of \(x\), the function is undefined. Let's look at another example in the function below. Instead of the independent variable, the dependent variable attains the value of \(+\infty\). It can also tend to \(-\infty\) for a finite value of \(x\).

Figure 3.- The Vertical Asymptote to a curve defined by \(y=f(x)\)

Here, the curve is approaching \(x=a\), but never touches it. Therefore, the vertical line \(x=a\) is a vertical asymptote. And again, if the value tends to something and never actually reaches it, it can be mathematically expressed as a limit that diverges:

$$ \lim_{x \to a} f(x)= +\infty \ \ \text{or} \ \ \lim_{x \to a} f(x)=-\infty$$

where the vertical asymptote is given by \(x=a\).

How to find Vertical Asymptotes of a function?

You saw how a vertical asymptote is defined mathematically, the function needs to approach some value \(x=a\) so that \(y\) approaches \(+\infty\) or \(-\infty\). You need to have such a value of \(a\) that the limits become undefined \((+\infty \ \text{or} \ -\infty)\).

Have you seen examples of this in your time studying math? You have! Dividing by zero is a huge no-no in math!

Thus, for every rational function, you must find a value of \(x\) such that the denominator will become \(0\). For instance,

\[y = \frac{5x}{x-3}.\]

is undefined when \(x=3\).

Some functions never equate to \(0\) for any real value of \(x\), for example, a class of exponential functions never attain the value \(0\). This implies that those functions don't have any vertical asymptotes. Let's look at an example to see what the above process looks like in practice:

Asymptotes don't always have to be exactly horizontal or vertical, they can be in any orientation. Asymptotes that make an acute angle with the x-axis are known as slant asymptotes.

Figure 4.- The Slant Asymptote to a curve defined by \(y=f(x)\)