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## Calculus, all content (2017 edition)

### Unit 1: Lesson 5

Formal definition of limits (epsilon-delta)# 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.

## Want to join the conversation?

- At ca3:20Sal 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)
- 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)

- Is it compulsory that when we approach any value of x in limits, the f(x) should be discontinuous at that point ?(5 votes)
- 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)

- What is the difference between X->A and x-> C ?(3 votes)
- they're the same thing essentially.

if you were to say x->a, we would look at values approaching from either side of a

like for x->c, we would look for the limit as x approaches c

really what i'm trying to say is that they're both just constants(14 votes)

- Wouldn't "L" here be equal to "f(c)"?(6 votes)
- 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)

- Epsilon is the greek Letter, right?(3 votes)
- Yes. In math it is usually used to represent an arbitrarily small (but not zero) quantity.(8 votes)

- 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)- 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)

- Why is it important to learn limits and calc?(3 votes)
- 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)

- 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)- 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)

- 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)
- You would just use normal limit rules that you use with most numbers. You should probably go to the level of precision that the real number has gone to, in this case, to the nearest millionth.(4 votes)

- why do we need to find a limit of a equation(2 votes)
- Limits of
*functions*are the base of calculus and are used to define what a derivative is later on. They're used in statistics as well.(2 votes)

## 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.