Limits at Infinity and 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 \(y=f(x)\)

Notice that the limit of the function converges to a finite value as \(x \rightarrow \infty\), which gives rise to the horizontal asymptote of the function.

It can be seen that the asymptote (the straight line) never crosses or even meets the curve itself. It may look like they meet each other at some point, but they only get ever closer.

As the value of \(x\) tends to \(+\infty\) the function tends to a certain value. As \(x\) approaches \(-\infty\), the value of the function starts to blow up, it approaches \(+\infty\).

For a curve described by the function \(y=f(x)\), if the function converges to a constant value, \(b\), then the equation of the horizontal asymptote is given by \(y=b\). In other words, \(y=f(x) \rightarrow b\) as \(x \rightarrow +\infty\).

The correct way to denote this is with limit notation.

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

All this equation means is that the limit of the function \(f(x)\) as \(x\) tends to infinite is \(b\).

Now, from this statement \(b\) could be an asymptote of \(f(x)\), but it might not. So, how can you find a horizontal asymptote?

How to find the Horizontal asymptotes of a function?

There is no universal algorithm or method for finding horizontal asymptotes, but there are some rules you can use to identify horizontal asymptotes of rational functions. If a rational function consists of a polynomial in both the numerator and the denominator, then you can find the asymptotes according to the following steps:

• 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:

Relationship between limits at infinity and Slant asymptotes

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 \(y=f(x)\)

The notion is very similar to that of the horizontal asymptotes, since one has to consider the limit of the function at infinity.

If you consider the limit of the function as \(x \rightarrow \infty\), the curve converges to a certain value which can be described by a straight line, which is the Slant Asymptote to the curve.

Let a function define a curve by \(y=f(x)\), then the slant asymptote to the curve is given by \(y=mx+b\), if and only if the below limits are satisfied:

$$ \lim_{x \to +\infty} \frac{f(x)}{x}=m \ \text{and} \ \lim_{x \to +\infty}|f(x)-mx|=b$$

If the above limits are not finite, then there are no slant asymptotes to the curve.

Limits at Infinity and Asymptotes Examples