The parts of the limit notation

The limit notation is a mathematical way of expressing the limit value that a function tends to as it approaches a value. It's a more subtle expression than stating the value of a function at a given point. It asks - and answers - the question: as the value of a function approaches a point, what will the limit value of the function be?

It's best understood by breaking down the parts of its notation. Let's take the following simple continuous function.

$$ \lim_{x \to 5} x^2 $$
Figure 1: Example of limit notation.

The preceding example is made up of the following parts.

  • The $x^2$ is the function we are investigating the limit of.
  • The $x$ is the variable.
  • $5$ is the value that we are approaching.

The expression poses the following question: what is the value of the $x^2$ function, as $x$ approaches $5$?

Compare that to what is the value of $x$ at $5$. We don't want the exact value at that exact point. We want to understand how the function behaves as it approaches that point.

Some examples

$$ \lim_{x \to 5} x^2 + 3 = 28 $$
Figure 2: What happens when x approaches 5?

In this preceding example, we want to know what value $x$ will tend to as it approaches $5$. Because this function is continuous, the limit will be the same as the value.

$$ \lim_{x \to 5} \frac {1} {x} = 0.2 $$
Figure 3: Limit notation on a discontinuous function.

Here things get more interesting because this function is not continuous throughout its domain, as when $x = 0$. In this preceding example, we want to know what value $x$ will tend to as it approaches $5$. Because this function is continuous, the limit will be the same as the value.

$$ \lim_{x \to \infty} \frac {1} {x} = 0 $$
Figure 4: The same function from figure 3, now approaching 4.

Staying with the same function, we can look at what happens to its value when $x$ approaches infinity. Said another way, we can look at the result when $x$ gets arbitrarily large.

$$ \lim_{x \to 0} \frac {1} {x} $$
Figure 5: An expression that yields undefined.

Finally, what happens when we take the same function, but $x$ approaches $0$? Because we can't divide by zero, the result of this expression is undefined.