Limit Operator

Limit notation annotated

The limit operator describes the result of an expression as a variable approaches a threshold value. The operator formalizes what mathematicians mean by “approach"[1]. For example, the limit operator typically appears in an expression like this:

In plain language, this means take the limit of the expression as the variable approaches the threshold value . In this case, the obvious approach is to plug in the value into the expression and see what happens.

After some simplification, we are left with zero divided by zero and, as you may recall, dividing by zero in math has undefined behavior. Unless we can invent something that makes the behavior of dividing by zero deterministic, we need another strategy. One thing we can try is to substitute values in for that get closer and closer to the threshold and see what happens.

So we can try substituting values like , and into the expression or values that approach from the other direction like , and and calculate the value of the expression. To our delight, a pattern emerges. It looks like the limit as approaches of this expression gets closer and closer to the value .

Here, we were able to reason about the limit by trying out different values with a computer. However, that is not always the case and there are other, better tools like calculus that help us compute them by hand[2]. And, again, the whole reason this operator exists is to make the answer to limit-like questions explicit and well-defined [1].

Plot of example limit

Since this page is incomplete, to learn more about limits I recommend two lessons from 3Blue1Brown’s calculus series.

  1. The formal definition of limits: (ε, δ) “epsilon-delta” definitions of limits
  2. Tools calculus gives us for evaluating limits: L’Hôpital’s rule


  1. (ε, δ) "epsilon delta" definitions of limits
    Grant Sanderson (3Blue1Brown)
  2. L'Hôpital's rule
    Grant Sanderson (3Blue1Brown)