In this lesson, our instructor Vincent Selhorst-Jones teaches the Formal Definition of a Limit. Youll learn the New Greek Letters, such as Delta and Epsilon, and figure out what it all means to you. Vincent will set up the ground work and then go over the Epsilon-Delta Part. Youll go on an adventure and then work through two fully explained examples.
Note: Very few students will have any use for what is in this lesson, let alone during a pre-calculus level course. Problems based on the formal definition of a limit are extremely rare in calculus class. This stuff won't come up in science classes,
and will only be necessary in high-level college math. Still, if you're interested in math for math's sake or know that you want to one day study higher level mathematics, check this lesson out. It's okay if it doesn't all make perfect sense today, but
seeing these ideas now will help you down the road when it starts mattering. Plus, it's fascinating stuff!
We need two new greek letters: Delta⇒ δ Epsilon⇒ ε
Formal Definition of a Limit: Let f be a function defined on some interval (a,b) of the real numbers, where a < b. Let c ∈ (a,b) and L be a real number. Then,
f(x) = L
means that for any real ε > 0, there exists some real δ > 0 such that for all x where 0 < |x−c| < δ, we have |f(x) − L| < ε.
The first half of the definition sets up the groundwork. The function is defined, there is a horizontal location we're approaching (c), and some vertical location the function goes towards (L).
The second half of the definition is the difficult part. It says that for any ε > 0 boundary around L, we can restrict our x to within some δ > 0 boundary of c to force f(x) to stay within the ε-boundary around L.
Check out the video to see some diagrams and more explanation of this idea. It's a confusing idea at first, and it takes awhile to make sense of it.
We can imagine a limit as a sort of never-ending debate between two people. One person gives hypothetical ε-boundaries, while the other person has to defend by giving a δ-boundary that would keep the function within the ε-boundary.
To formally prove a limit exists, we must show that for any ε > 0, there will always be some δ > 0. This normally takes the form of creating a formula for δ based off ε, because if we can create such a formula, then we will
have shown δ will exist no matter what ε is chosen.
Formal Definition of a Limit
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.