One-sided limit

Usually we work out a two-sided limit, that is, we find what value the function values approach both from the left and from the right. But we can also look at a one-sided limit – only from the left or only from the right.

One-sided limits

Let’s take some function f(x) and some point x₀ at which we want to work out the limit. Normally we would write:

$$\underset{x\to x_0}{lim}f(x)=?$$

But we can approach the point x₀ from one side only – either from the left or from the right. This is written with a minus or plus sign in the superscript. So “x tending to 7 from the left” and “x tending to 10 from the right” would be written like this:

$$\begin{array}{rcl}\underset{x\to 7^{-}}{lim}f(x)& =& L_1\\ \underset{x\to {10}^{+}}{lim}f(x)& =& L_2\end{array}$$

We modify the definition simply by taking only the left or right neighbourhood of the accumulation point instead of the whole neighbourhood. The modified definition of the limit from the right:

$$(\mathrm{\forall }ϵ>0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in \mathrm{dom}(f))((x_0<x<x_0+\delta )\Rightarrow |f(x)-L|<ϵ).$$

Notice that in the implication we take only those x that are greater than x₀ – that is, those x that are to the right of the number x₀ on the number line. The definition of the limit from the left:

$$(\mathrm{\forall }ϵ>0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in \mathrm{dom}(f))((x_0-\delta <x<x_0)\Rightarrow |f(x)-L|<ϵ).$$

In general the left-hand and right-hand limits need not be equal. Several cases can occur: the function has both a left-hand and a right-hand limit at the point, the function has only one of them, or it may have neither.

It holds that the function f has a limit at the point x₀ if and only if it has both a left-hand and a right-hand limit at that point and the two limits are equal. If it has both one-sided limits but they are different, then the function has no limit at that point – it has only one-sided limits.

Let’s demonstrate all this on the simple function f(x) = 1/x. Here is the graph:

A trivial case is the limit as x tends to one. At the point one the function is continuous and defined, and it is altogether a nice example. From both the left and the right the function value approaches one, so the limit will be one too. But what if we want to work out the limit as x tends to zero?

$$\underset{x\to 0}{lim}\frac{1}{x}=?$$

Now this is an interesting example. Notice that we are trying to find a limit at a point that is not in the domain of the function. If we substitute zero into the function, we get an invalid expression (you can’t divide by zero!). That doesn’t bother us at all, because the definition talks about an accumulation point, and zero is an accumulation point of the domain, since in every neighbourhood of it we find some element of the domain of the function.

If we look at the graph and start approaching zero from the left, we can see that the function value keeps decreasing. Examples:

$$\begin{array}{rcl}f(-10)& =& -1/10\\ f(-1)& =& -1\\ f(-\frac{1}{2})& =& -2\\ f(-1/10)& =& -10\end{array}$$

If we took a negative number that is “very” close to zero, say −0.000 001, we would get a “very” small number as the function value: −1,000,000.

The closer we get to zero from the left, the closer the function value gets to minus infinity. On the other hand, we can see that if we approach zero from the right, the function value grows. The closer we get to zero from the right, the closer the function value gets to infinity. How do we resolve this? We simply say that the function has no limit at zero. It has only two different one-sided limits:

$$\begin{array}{rcl}\underset{x\to 0^{-}}{lim}\frac{1}{x}& =& -\mathrm{\infty }\\ \underset{x\to 0^{+}}{lim}\frac{1}{x}& =& +\mathrm{\infty }\end{array}$$