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## Calculus, all content (2017 edition)

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

Lesson 1: Limits introduction

# Limits intro

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.A (LO)
,
LIM‑1.A.1 (EK)
,
LIM‑1.B (LO)
,
LIM‑1.B.1 (EK)
A limit tells us the value that a function approaches as that function’s inputs get closer and closer to some number. The idea of a limit is the basis of all calculus. Created by Sal Khan.

## Want to join the conversation?

• Does anyone know where i can find out about practical uses for calculus?
• The amount of practical uses for calculus are incredibly numerous, it features in many different aspects of life from Finance to Life Sciences to Engineering to Physics. It's hard to point to a place where you could go to find out about the practical uses of calculus, because you could go almost anywhere. I recommend doing a quick Google search and you'll find limitless (pardon the pun) examples.
• Do one-sided limits count as a real limit or is it just a concept that is really never applied?
• It does get applied in finding real limits sometimes, but it is not usually a "real limit" itself. For instance, let f be the function such that f(x) is x rounded to the nearest integer. What is the limit of f(x) as x approaches 0.5? Well, there isn't one, and the reason is that even though the left-hand limit and the right-hand limit both exist, they aren't equal to each other.

Of course, if a function is defined on an interval and you're trying to find the limit of the function as the value approaches one endpoint of the interval, then the only thing that makes sense is the one-sided limit, since the function isn't defined "on the other side".
• Are there any textbooks that go along with these lessons? It would be great to have some exercises to go along with the videos.
• There are many many books about math, but none will go along with the videos.
The idea behind Khan Academy is also to not use textbooks and rather teach by video, but for everyone and free!
• What exactly is definition of Limit?
• The strictest definition of a limit is as follows: Say Aₓ is a series. If there exists a real number L that for any positive value Ԑ (epsilon), no matter how small, there exists a natural number X, such that { |Aₓ - L| < Ԑ, as long as x > X }, then we say A is limited by L, or L is the limit of A, written as lim (x→∞) A = L.

This is usually what is called the Ԑ - N definition of a limit. (I replaced the n's and N's in the equations with x's and X's, because I couldn't find a symbol for subscript n).
• 0/0 seems like it should equal 0. Why doesn't it?
• Nothing can be divided by zero, not even zero itself.
• Would that mean, if you had the answer 2/0 that would come out as undefined right? since x/0 is undefined :( just want to clarify
• Anything divided by zero is undefined, yes.
• What is the difference between calculus and other forms of maths like arithmetic, geometry, algebra, i.e., what special about calculus over these(i see lot of basic maths are used in calculus, are these structured in our school level maths to learn calculus!!).
• Elementary calculus may be described as a study of real-valued functions on the real line. One divides these functions into different classes depending on their properties. Examples of such classes are the continuous functions, the differentiable functions, the integrable functions, etc.

Some insight will reveal that this process of grouping functions into classes is an attempt to categorize functions with respect to how "smooth" or "well-behaved" they are. For instance, an integrable function may be less smooth (in some appropriate sense) than a continuous function, which may be less smooth than a differentiable function, which may be less smooth than a twice differentiable function, and so on.

If one knows that a function `ƒ` is continuous, what else can you say about `ƒ`? The intermediate value theorem, the extreme value theorem, and so on, are examples of theorems describing further properties enjoyed by continuous functions. One should regard these theorems as descriptions of the various classes.

And then there is, of course, the computational aspect. Elementary calculus is also largely concerned with such questions as how does one compute the derivative of a differentiable function? How does one compute the integral of an integrable function? Here there are many techniques to be mastered, e.g., the product rule, the chain rule, integration by parts, change of variable in an integral.

Some calculus courses focus most on the computational aspects, some more on the theoretical aspects, and others tend to focus on both.

The reason you see a lot of, say, algebra in calculus, is because many of the definitions in the subject are based on the algebraic structure of the real line. Many aspects of calculus also have geometric interpretations in terms of areas, slopes, tangent lines, etc.
• I'm not quite sure I understand the full nature of the limit, or at least how taking the limit is any different than solving for Y. I understand that if a function is undefined at say, 3, that it cannot be solved at 3. However, wouldn't taking the limit as X approaches 3.00001 or 2.99999 be the same as solving for X at these points? I'm sure I'm missing something.
• A limit is a method of determining what it looks like the function "ought to be" at a particular point based on what the function is doing as you get close to that point. If you have a continuous function, then this limit will be the same thing as the actual value of the function at that point. In fact, that is one way of defining a continuous function:
A continuous function is one where
f(c) = lim x→c⁻ f(x) = lim x→c⁺ f(x) for all values of c within the domain.

But, suppose that there is something unusual that happens with the function at a particular point. If the function is not continuous, even if it is defined, at a particular point, then the limit will not necessarily be the same value as the actual function.
Suppose we have the function:
`f(x) = 2x, where x≠3, and 200, where x=3`
So, this function has a discontinuity at x=3.
Thus, f(3) = 200
But lim x→3 f(x) = 6, because, it looks like the function ought to be 6 when you get close to x=3, even though the actual function is different.

The other thing limits are good for is finding values where it is impossible to actually calculate the real function's value -- very often involving what happens when x is ±∞. Since ∞ is not a number, you cannot plug it in and solve the problem. But you can use limits to see what the function ought be be if you could do that.
lim x→+∞ (2x² + 5555x +2450) / (3x²)
We can determine this limit by seeing what f(x) equals as we get really large values of x.
f(10) = 194
f(10⁴) ≈ 0.8518
f(10⁶) ≈ 0.6685185
f(10¹⁰) ≈ 0.66666685
f(10²⁰) ≈ 0.666666666666666685
Quite clearly as x gets large and larger, this function is getting closer to ⅔, so the limit is ⅔.

Note: using l'Hopital's Rule and other methods, we can exactly calculate limits such as these, so we don't have to go through the effort of checking like this.