The precise definition of the limit EXPLAINED! (KristaKingMath)

The precise definition of the limit, also called the epsilon-delta definition, is the proof of the concept of the limit. It proves the limit because it shows how, as you move closer and closer to a particular value of x, the value of the function will move closer and closer to a particular value of y.

If you want to use technical terms, what you're doing is limiting the value of delta, which is the distance you can move away from x, in order to limit the value of epsilon, which is the distance you can move away from the corresponding value along the y-axis.

To sum it up, if you found the limit some other way using calculus (like with substitution, factoring, conjugate method, L'Hospital's rule, etc.), this is the theorem that would let you prove that the value of the limit you already found is actually legitimate. It tells you why the calculus works how it does.

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”

[nicholasberry]

You are literally the only person I found who bothered to explain what delta and apsilom actually was, thank you so much

[kainny]

Taking calculus in college, 1 hour video from my teacher is nothing compared to this, thanks!

[debajitpatowary]

This is a common line you've heard millions of times, but i wanna say the line: Thank You. I've not seen any simpler explanation of Epsilon-delta than this one and I'm not confused anymore. A Big Thank You and Congratulations!

[donnydorito]

I learned this yesterday in class and the teacher lost me as soon as he said Epsilon/Delta. This was a fantastic explanation which I understood completely. Thanks!

[RubiMercuri]

omg i watched about 5 videos to try to understand this definition and it finally clicked completely after watching yours! you explained this so well!! thank you so muuuuch!

[Mravsfan]

As a math tutor, I think I'll find this video helpful for helping students. Thanks for sharing!

[kingbeauregard]

Yours is the best explanation of this I have thus far found, as I keep looking trying to understand. (Fortunately I am long out of college and am doing this just for fun, for certain definitions of "fun").

Here's what I've managed to mash into my brain:

- A limit for a given y value exists at a given x value, if you can say that, as you get closer and closer to that y value, you also get closer and closer to that x value. That's intuitive from a human brain perspective; the trick is to express that with mathematical rigor and precision.

- A good way to do "epsilon implies delta" is to find a function that relates the two. It's not strictly required, but it's the simplest way to prove all values of epsilon and delta, and it works well for non-pathological functions, so let's try to do that. If you can demonstrate the relationship, then you've got a couple of nested inequalities: epsilon is the outer bound of whatever relationship you've set up with delta, which in turn is the outer bound of how far you're straying from your target x.

- The epsilon / delta thing can break if you're trying to be jerk about it, but limits assume you're not trying to be a jerk. Like if you were trying to do the limit of sin(x) as x approaches 0, if you try to set up your delta and epsilon and say "a-HA! but it doesn't work if you evaluate epsilon and delta at x = 23 pi", no, that's just you being a jerk. When you're near x = 0, it works fine. As in all things, don't be a jerk.

Sooo. I'm trying to prove the existence the limit of sin(x) as x approaches 0. What I end up with is that epsilon > |sin(delta)| ; is that a correct result? I THINK I have demonstrated a relationship between epsilon and delta, which means that any little distance I stray from x = 0 (less than an arbitrary delta) will mean that my function will stray from sin(0) by less than epsilon (which equals |sin(delta)| ). I can make delta crazy small, which means epsilon can get crazy small too, which means that I can get epsilon down to effectively zero (though not exactly zero). And that's the limit.

[maxdegreat]

this is the first video that i actually understood

[girlgerber]

Never taken a Calculus class in my life....great explanation. Thanks for sharing.

[tanisha]

thank you so much literally no one else explained what epsilom and delta is. you have helped me a lot!

[mariocraft]

Thanks a million! I remember this class, and I remember being extremely confused. I just did my homework in about 15 minutes for this section after watching this video. Absolutely amazing

[georgepolasky]

Krista, as I’ve said time and time again, you are brilliant, a great, great teacher, and a wonderfully nice person, all simultaneously. You hit the Trifecta. Hardly anyone can do that. You’re the best. ☺️❤️👏👍

[cryomnia]

I just started Calculus this semester and honestly, this section really confused me more than I was expecting. Thanks so much for clearing this up, incredibly helpful!

[strangertodeath]

I was having a hard time with calculus until I found your videos. I owe you so much and I have to Thank you.

[ColleenDickens-wi]

Your explanations are always so precise!

[nucleus]

I don't know you but I love you lol. You teach really well!!!

[abuibraheem]

Thanks ten million times for this explanation, I was looking in the internet for days, you are amazing in the way you convey the lesson .

[neilpeteson]

U just saved me. Cant even explain how confused i was after this was covered in my lecture. THANK YOU

[lidyakumar]

I'm glad I clicked on this video it made so much more sense with the explanations! Thank you so much!

[tichaonamarariromba]

You are best krista king I passed all many mathematics modules through yr tutorials