Limit Operator

Limit notation annotated

The limit operator describes the result of an expression as a variable approaches a value. The operator is used in calculus to formalize what mathematicians mean by “approach"[1]. For example, limits are used in the definition of derivatives and typically appear in expressions like the one below.

In this case, after some algebra the fraction simplifies and the question becomes almost trivial. What is the value of as approaches ? Substituting values in for that approach the value of the expression clearly approaches .

So why does the limit operator exist and what role does it play in calculus? Consider another example, shown below, where if you go straight to substituting in for into the expression you run into a problem: zero by zero is an undefined operation in math.

One way to reason about limits is to simply plug in values that get closer and closer to the threshold value. Substituting values in for that get closer and closer to the value the pattern emerges.

So the whole expression approaches a very real and concrete number .

There are better ways to justify this particular result, but the point is the limit operator helps describes the result of problems that appear paradoxical or undefined when, in reality, they have meaningful and concrete results[2]. That’s not to say there always is an answer, sometimes a limit does not exist. Instead, the operator makes the answer explicit and well defined.


  1. (ε, δ) "epsilon delta" definitions of limits
    Grant Sanderson (3Blue1Brown)
  2. Essence of Calculus Series
    Grant Sanderson (3Blue1Brown)