Calculus Limits Epsilon delta definition of limits Limit intuition review Building the idea of epsilon-delta definition Epsilon-delta definition of limits Proving a limit using epsilon-delta definition Limits to define continuity Additional content Epsilon Delta Limit Definition 1 Epsilon Delta Limit Definition 2 Limit intuition review Back Limit intuition review ⇐ Use this menu to view and help create subtitles for this video in many different languages. You'll probably want to hide YouTube's captions if using these subtitles. Let's review our intuition of what a limit even is. So, let me draw some axes here. Let's say this is my y axis, so try to draw my vertical line That right over there is my y axis and let's say this is my x axis. I'll focus on the first quadrant, although I don't have to... Let's say this right over here is my x axis and let me draw a function. So let's say my function looks something like that. It could look like anything suitable, so this is y is equal to f of x and just for the sake of conceptual understanding I'm going to say it's not defined at a point. I didn't have to do this, you could find the limit as X approaches a point where the function actually is defined, but it becomes that much more interesting, at least for me where you start to understand where a limit might be relevant when a function is not defined at some point. So, the way I've drawn it this function is not defined when x is equal to c Now, the way that we've though about a limit is, what does f(x) approach as x approaches c? So, let's think about that a little bit. When x is a reasonable bit lower than c, f(x) for our function that we just drew is right over here y is equal to f(x) when x gets a little bit closer, then our f(x) is right over there when x gets even closer; It's really almost at c but not quite c then our f(x) is right over here and the way we see it, we see that our f(x) seems to be that as x gets closer to c it looks like our f(x) is getting closer to some value. right over there, I'll even draw it with a more solid line. and that was only the case when x was getting closer to c from the left, from values less than c. But what happens when we get closer and closer to c from values of x that are larger than c, well when x is over here, f(x) is over here. That's what f(x) is when x is over there, x gets a little bit closer to c our f(x) is right over there, when x is just very slightly larger than c, then our f(x) is right over there and you see once again it seems to be approaching that same value and we call that value, the value f(x) seems to be approaching as x approaches c, we call that value L, or the limit and we'd call the limit, we don't have to call it L all the time. and the way that we would write that mathematically is, the limit of f(x) as x approaches c is equal to L and this is a fine conceptual understanding of limits and it really will take you very far, and you are ready to progress and start taking a lot of limits. But this isn't a very mathematically rigourous definition of limits. This sets us up for the intuition, in the next few videos we will introduce a mathematically rigourous definition of limits. That will allow us to do things. Like prove that the limit as x approaches c truly, is equal to L. Questions Tips & Feedback Be specific, and indicate a time in the video: At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger? Have something that's not a question about this content? Post a tip or feedback General discussion about the site Report a technical problem with the site Request a video or feature This discussion area is not meant for answering homework questions. Formatting tips Cancel or ( total) Share a tip When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831... Suggest a fix At 2:33, Sal says "double bonds" but should say "single bonds." Have something that's not a tip or feedback about this content? Ask a question General discussion about the site Report a technical problem with the site Request a video or feature This discussion area is not meant for answering homework questions. Formatting tips Cancel or Discuss the site For general discussions about Khan Academy, visit our Reddit discussion page. Flag inappropriate posts Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians. abuse disrespectful or offensive an advertisement not helpful low quality not about the video topic soliciting votes or seeking badges a homework question a duplicate answer repeatedly making the same post wrong category a tip or feedback in Questions a question in Tips & Feedback an answer that should be its own question about the site a question about Khan Academy (Visit our FAQ) a post about badges a technical problem with the site (Report a problem) a request for videos or features
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