Jorge Valle

Jorge Valle

Jorge Valle is a front end developer with a particular passion for, and expertise in, JavaScript and user interfaces. Lately, he's also been diving into machine learning.

Understanding limit notation

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.


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