Question

Teach me calc please!

Answer 1

Calculus almost always begins with the lesson on "Limits"
And Limits..almost always begins with an intuitive approach. Got that so far?

Answer 2

Sorry my connection sucks a__, I was like wt_

Anyway, sorry continue please

Answer 3

Let's begin with notation:
\[\Large \lim_{x\rightarrow a}f(x) = L\]

Answer 4

This is read as "The limit of f(x) as x approaches a is L"

Answer 5

So it is limited to the a, but it is never actually a, wio told me that.

Answer 6

Not "never actually a" as much it doesn't HAVE to be a.

Let's have a simple example.
\[\Large f(x) = x + 2\]

Answer 7

Find the limit of f(x) as x approaches 1.

Answer 8

An intuitive approach would be to get values really close to 1, but not exactly 1.
Things like, 0.5, 1.5, 0.75, 1.25, etc
and evaluate f at those values.

Answer 9

By plugging in these x values?

Answer 10

Yup.
Let's have a table
\[\Large \begin{matrix}x&f(x)\\0.5&2.5\\0.75&2.75\\1.25&3.25\\1.5&3.5\end{matrix}\]

Answer 11

I see

Answer 12

As you can see, the closer x gets to 1, the closer f gets to 3.

Answer 13

Yes

Answer 14

Intuitively, we conclude that
\[\Large \lim_{x\rightarrow 1}(x+2)=3\]
With me so far?

Answer 15

yes

Answer 16

but it is supposed to only approach 3, but not be equal to 3, right?

Answer 17

It's only supposed to approach 3, it *doesn't HAVE TO* be equal to 3 at that point.
But it could.

As you can see, not only is the limit of f as x approaches 1 equal to 3, but
f(1) = 3 as well.

Good. But this is intuitive, so therefore not conclusive. Is there any way to properly define a limit so that it doesn't rely on intuition/guesswork?

Answer 18

There is a proper definition of Limits, did wio expound on that?

Answer 19

Kind of, but I am not sure whether I can exactly do those yet, can you explain that again?

Answer 20

It's just a definition, it's not strictly needed, unless you're proving. Anyway
\[\Large \lim_{x\rightarrow a}f(x) = L\]\[\Large \iff\]
\[\Large \text{for any } \varepsilon>0 , \text{there exists } \delta > 0 \text{ such that} \]
\[\Large \text{whenever } |x-a|<\delta \]\[\Large \left|f(x) -L\right|<\varepsilon \]

Answer 21

Does that look daunting? :D

Answer 22

what does that cursive b stand for?

Answer 23

It's the lowercase Greek letter, delta.
It represents a positive value.

Answer 24

So does epsilon \(\varepsilon \)

Answer 25

And what's the difference b/w cursive e and cursive b?

Answer 26

They're both positive values. This definition simply means that they need not be the same.

Answer 27

Ok

Answer 28

Okay, I take it you didn't fully understand the definition yet? :D

Answer 29

and b and e, have to be positive integers?

Answer 30

integers? hardly.
They're understood to be really small, like, infinitesimal...

Answer 31

So there don't have to be, but most likely are decimals?

Answer 32

Yes. They're arbitrary. Let me explain the definition in plain English (as best I could) and maybe you'll have a clearer picture of what's really happening, right? :D

Answer 33

k

Answer 34

Calculus is my heart.

Answer 35

@felavin, sorry to ask, but please don't interrupt in the future like this, OK?

Answer 36

For any \(\varepsilon > 0\) at this point, we're just picking a random distance.
there is a \(\delta > 0\) another distance, but fact that the phrase "there exists" is placed before it means that more often than not, the value of \(\delta\) depends on the value of \(\varepsilon \) selected.

Answer 37

Now, the fun part... whenever \(\large |x-a| < \delta\) any idea what this means?

Answer 38

It means, whenever x is no more than \(\delta\) units away from a... IE, when the distance of x from a is no more than delta....

Answer 39

Didn't understand the last part, "when the distance of x from a is no more than delta...."
(sorry)

Answer 40

Newton and Leibniz had no conception of a limit, and they're the people who invented the subject. Although limits are nice, they often conceptually miss the point and I suggest you find a more intuitive introduction to calculus @SolomonZelman rather than waste your time with proofs.

Answer 41

Okay, the distance between two numbers is basically their difference, right?

Answer 42

Like, the distance between 4 and -3 is 7, since 4 - (-3) = 7.

Answer 43

@Kainui, and all of that just span my thread, nothing more...

Answer 44

and what is delta?

Answer 45

Kai does have a point... this proof is only to show you the 'beauty' of calculus... but not quite its practical side yet.
If you wish, I can skip the theoretical bits.

Answer 46

I'm trying to help you understand that you're wasting your time on this definition of a limit right now haha. I seriously suggest you skip it, integrals and derivatives are beautiful and have a utility whether or not you know a rigorous definition of a limit or not.

Answer 47

If that is not necessarily for my initial question, than please do as you feel like.

Answer 48

Okay, good. Just accept that a limit of a function is what that function approaches as x goes to some number. If you got that, then you're past lesson one.. this thread is getting laggy, mind making a new one? :D

Answer 49

I see what you want, you do deserve more....

Answer 50

No, that's not it at all.
Feel free to stress in your new thread that I'm not to be given any medals ^_^

Answer 51

Yeah didn't mean to sort of bust in here, I've just seen too many people with the ability to take derivatives and integrals all day long without actually knowing what they're doing.