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Formal definition of limits Part 1: intuition review

A quick reminder of what limits are, to set up for the formal definition of a limit. Created by Sal Khan.

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  • male robot hal style avatar for user KEVIN
    At ca Sal mentions a "rigorous" defintion of limits. In regards to mathematics in general, what is the threshold for a definition being rigorous? When do we know that a certain level of rigor has been achieved? Thanks for any insight.
    (2 votes)
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    • piceratops tree style avatar for user preethvijay17
      The definition of the word "rigorous" is basically strict, extremely accurate, or formal. Thus, a rigorous definition in general (and this can also be applied to mathematics) is simply a formal, thorough definition of a certain concept or idea
      (26 votes)
  • aqualine seedling style avatar for user vishwas sethia
    Is it compulsory that when we approach any value of x in limits, the f(x) should be discontinuous at that point ?
    (5 votes)
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    • aqualine ultimate style avatar for user Harshit
      Not necessarily. But we mostly use limits where the function is undefined (ie discontinuous) to understand what the graph would look like very close to that point.

      We can take limit at a place where f(x) is defined eg f(x)=x^2 an put a limit x-->3 here the ans will be same as f(3)=9(ie x is approaching 9 at f(3)) so its not that useful for a defined value of f(x).

      But for an function like that given in "limits by factoring" video where f(x)=(x+3)(x-2)/(x-2) func is undefined at x=2 so we will use limit to know what value does the graph gives nearby f(2). Here if we directly put x=2 then func. will become 0/0 and you can't just cancel 0/0. But if we put limit x-->2 (x-2) will become something either just less than 0 or just more than 0 so we can cancel them out like {(-0.00001)/(-0.00001)=1} now our eq. becomes (x+3) so if we put x=something very close to 2, f(x) will become something very close to 5

      [It is the value of f(x) we would have got if x was defined for 2]

      PS:I hope this helps and not confuses you more
      (4 votes)
  • blobby green style avatar for user vercillob
    What is the difference between X->A and x-> C ?
    (3 votes)
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  • leaf orange style avatar for user Student
    Wouldn't "L" here be equal to "f(c)"?
    (6 votes)
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    • blobby green style avatar for user Tom
      No, because in this example f(x) is specifically undefined at c. So while the limit of f(x) as x approaches c is L, f(c) does not actually exist.

      If you go back and watch the beginning of the video again, he draws the function with f(c) in it but then erases a small space and puts an empty circle in that spot to show that the function is undefined there. This is to show that the function doesn't need to exist at the point at which we take a limit, though it can.
      (2 votes)
  • ohnoes default style avatar for user Brian Lee
    Epsilon is the greek Letter, right?
    (3 votes)
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  • blobby green style avatar for user Bella Lee
    If you were given a function...
    f(x) = l x+ 7 l /x+7, and you are just expected to find the x value (if any) at which f is not continuous, (I know how to do this.. I think haha) you can't just find the limit of that function just with the given information because you are not given what x is approaching to, correct? So you can only get the limit if you are given what x approaches to? Thank you! :-)
    (4 votes)
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    • duskpin ultimate style avatar for user Patrick Stetz
      It is possible to find the point of discontinuity even if all you know is that
      f(x) = I x + 7 I / (x + 7)
      For this example there will be no value of f(x) when the denominator is zero (when x = -7)
      There will be a discontinuity at x = -7 because f(x) doesn't exist there
      I hope this helps!
      (6 votes)
  • orange juice squid orange style avatar for user jonah.yoshida
    Why is it important to learn limits and calc?
    (3 votes)
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    • blobby green style avatar for user Creeksider
      We learn limits because they're essential to learning calculus, and we learn calculus because . . . so many reasons! It vastly increases the number of math problems you can solve, it's one of the great achievements in human thought, it's a challenge that will improve your mind, it's inherently beautiful . . .
      (6 votes)
  • primosaur tree style avatar for user M
    I have a question based the topic.
    The opposite sides of a cyclic quadrilateral are supplementary. What does this this proposition become in a limit when two angular points coincide?
    This question is from the book DIFFERENTIAL CALCULUS FOR BEGINNERS BY JOSEPH EDWARDS
    Kindly answer as soon as possible as I need this information for an upcoming test.
    (4 votes)
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    • leaf red style avatar for user Bean Jaudrillard
      Assuming the quadrilateral stays cyclic as two angular points (say A and B) coincide, the proposition clearly holds as A --> B, though at B it becomes meaningless because the quadrilateral becomes an inscribed triangle and the value of A is undefined. There is also no notion of approaching B "from the other side" (as in a left-hand limit) and so the limit A --> B doesn't exist.
      (2 votes)
  • leaf red style avatar for user DannyQ
    I understand how to calculate the the limit when it's a whole number or infinity but how would determine it for a real value such as 1.254325? Also, to what level of precision should you go to?
    (3 votes)
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  • spunky sam red style avatar for user msk basha
    why do we need to find a limit of a equation
    (2 votes)
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Video transcript

Let's review our intuition of what a limit even is. So let me draw some axes here. So let's say this is my y-axis, so try to draw a vertical line. So that right over there is my y-axis. And then let's say this is my x-axis. I'll focus on the first quadrant, although I don't have to. So let's say this right over here is my x-axis. And then let me draw a function. So let's say my function looks something like that, could look like anything, but that seems suitable. So this is the function 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 can 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, or you start to understand why a limit might be relevant where 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 thought about a limit is what does f of 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 of x, for our function that we just drew, is right over here. That's what f of x is going to be equal. y is equal to f of x. When x gets a little bit closer, then our f of x is right over there. When x gets even closer, maybe really almost at c, but not quite at c, then our f of x is right over here. And the way we see it, we see that our f of x seems to be-- as x gets closer and closer to c it looks like our f of x is getting closer and closer to some value right over there. Maybe I'll draw it with a more solid line. And that was actually only the case when x was getting closer to c from the left, from values of x less than c. But what happens as we get closer and closer to c from values of x that are larger than c? Well, when x is over here, f of x is right over here. And so that's what f of x is, is right over there. When x gets a little bit closer to c, our f of x is right over there. When x is just very slightly larger than c, then our f of x is right over there. And you see, once again, it seems to be approaching that same value. And we call that value, that value that f of x seems to be approaching as x approaches c, we call that value L, or the limit. And so the way we would denote it is we would call that the limit. We don't have to call it L all the time, but it is referred to as the limit. And the way that we would kind of write that mathematically is we would say the limit of f of x as x approaches c is equal to L. And this is a fine conceptual understanding of limits, and it really will take you pretty far, and you're ready to progress and start thinking about taking a lot of limits. But this isn't a very mathematically-rigorous definition of limits. And so this sets us up for the intuition. In the next few videos, we will introduce a mathematically-rigorous definition of limits that will allow us to do things like prove that the limit as x approaches c truly is equal to L.