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## Calculus 1

### Course: Calculus 1>Unit 1

Lesson 4: 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 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.
• 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
• Wouldn't "L" here be equal to "f(c)"?
• 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.
• Is it compulsory that when we approach any value of x in limits, the f(x) should be discontinuous at that point ?
• 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
• What is the difference between X->A and x-> C ?
• 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
• Epsilon is the greek Letter, right?
• Yes. In math it is usually used to represent an arbitrarily small (but not zero) quantity.
• 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! :-)
• 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!
• Why is it important to learn limits and calc?
• 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 . . .
• 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.
• 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.