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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 1

Lesson 1: Limits introduction

# Introduction to limits (old)

An older video of Sal introducing the notion of a limit. Created by Sal Khan.

## Want to join the conversation?

• What exactly is the definition of a Limit?
• as your graph aproaches some value of x, what is its height (y value)
• Hi there,, since we know the result is 0 whenever 0 is multiplied by any number. What happens when 0 is multiplied by an undefined number??????
• Help ! I'm not sure that I completely understand one point in this video. Starting at Sal says, quote: " In this case the limit as x approaches 2 is also equal to 4; this is interesting because in this case the limit as x approaches 2 of f(x) does not equal f(2).
Does this mean that when stating the limits of this function we must state BOTH that:
lim f(x) as x approaches 2 = 4
and lim f(x) as x approaches 2 # f(2) ?
Everything else is clear.
• Ok, Sal started the video by talking about the function f(x)=x^2. Then he added a piecewise function f(x) = x^2 when x does not equal 2, f(x) = 3 when x =2. In both cases the limit as x approaches 2 is 4. Only in the first case where the function is f(x) = x^2 is f(2) = 4.

This is an important point about limits. The function does not have to be defined at the point the limit is being evaluated at for the limit to exist. We see this in the second function. The second function is not continuous at the point x=2 since f(2) does not equal 4. The limit is still 4 since as we get closer and closer to x=2 the value of f(x) gets closer and closer to 4. Play with this with your calculator and you'll see what's happening.

I hope I'm not confusing you more. Just think about how we phrase a limit, that is, the limit of f(x) as "x approaches a" is equal to y. The "x approaches a" tells us that we aren't considering what f(x) actually is at "a" but what f(x) is as we evaluate numbers closer and closer to "a". All of this is important in calculus since it forms the definition of a continuous function.
• Why do people say, "Find the LIMIT of a function as it approaches some number"? Why can't we just say, "Find the value of the function as it approaches this number"? What does it mean by taking the "limit" of a function?
• The limit may or may not be the same thing as the value of the function.
The limit is what it LOOKS LIKE the function ought to be at a particular point based on what the function is doing very close to that point. If the function makes some sudden change at that particular point or if the function is undefined at that point, then the limit will be different than the value of the function.
• I still don't understand after watching the video multiple times. Why is there a limit? What does it mean "as the limit of x approaches 0 or 2"? Why? Why? Why? I'm so confused.
• The concept is important because it explains to us about the function's behavior on an x-y plane. The concept of limits may not seem important at this time, but in jobs, especially physics of motion, deals a lot with limits. If you were to take a rocket, for example, and want to know where it is at a certain time (x), you can use limits to identify where it is at. For example, as the limit of x approaches 3 seconds, the rocket is approaches 50 kilometers.

A strong background in Algebra II and PreCalculus will solidify your knowledge of limits. Those subjects explain the basics of limits, and Calculus will show you some application of those limits (in continuity, rate of change, velocity.....and so much more).
• does the hole represent discontinuity in the function?
• If you are speaking of an empty circle on the graph, then, yes, that is how a discontinuity is depicted. Either the function is not defined at that point or else there is a sudden, discontinuous, jump in what the value of the function is at that point.
• Hello, can someone please give me some possible applications of limits in physics, engineering, etc. Just curious.
• Some specifc examples might also be right here in mathematics. If you remember from conic sections we actually use limits to find the assympthotes of the hyperbola, also you can use limits to define other cool stuff. For example, imagine that suddenly the formula to get the area of a circle has been removed from all knowledge, texbooks, etc... and you want to calculate the area of a circle. You might say "G, it's imposible!" but actually limits can help with this problem. If you take a polygon, let's say a square, you can put it inside the circle and the area of the square is going to be somehow close to the area of the circle, but still very far off from the real value. However as you add more and more sides to the polygon (imagine an hexagon inside of the circle) then the area of the polygon APPROCHES the area of the circle more and more. Now let's say "S" denotes the number of sides of the polygon, then you can define a function in wich to determine quite precisely the area of the polygon in terms of "S". So you could state that as "S" (The number of sides of the polygon) approches infinity, then the area of the polygon approches or is basically the same as the area of the circle. Pretty cool Uhh?

Hope this gives you an insight to what you can do with limits and really encourage you to keep learning about this topic in particular.

P.S. If you'd like a more solid explanation about the math involved to the problem of the circle and the polygon feel free to tell me and i'll work out the math.
• What is the purpose of limits? What do they help you find out about a function?
• let's say a function f(x) is undefined at a point, assume it to be 0. But you want to find the value very very very close to zero, in that case limits are useful.
Limits are also used to derive the derivative of a function
• hii..my question is that why the value of 0! and 1! is same.....till now i ask to many teacher but no one told me the reason....so please give me answer as soon as possible...because i very excite to know about it...