Question

Teach me calc 2!

Answer 1

They will come again, I had the same with wio trying to help me.

Answer 2

Continued from
http://openstudy.com/study#/updates/5263f60ae4b029b030d980d4

Answer 3

Anyway, out of limits, onto the concept of continuity.
This one's fairly easy to grasp.

Answer 4

You might have realised that that not all limits of functions are the functions evaluated at the limit point, right?

Answer 5

I missed what you said; I am not smart enough to get that phrase....

Answer 6

I mean, \[\Large \lim_{x\rightarrow a}f(x)\]
is not necessarily equal to \[\Large f(a)\]

Answer 7

You want an example? :)

Answer 8

I see,

Answer 9

But please example, that would be good 2

Answer 10

Consider\[\Large f(x) = \frac{x^2-4}{x-2}\]
and try finding \[\Large \lim_{x\rightarrow 2}f(x)\]
*if you can*

Answer 11

x+2

Answer 12

the limit, find it.

Answer 13

Suffice it to say, the limit of the function as x goes to 2 is 4, right?
Because, as you have seen, the function is almost identical to x+2.

Answer 14

However, there is no such f(2)
It will not be defined, as it results in a 0/0.
Hence, vacuously \[\Large\lim_{x\rightarrow2}f(x)\ne f(2) \]

Answer 15

I don't get why is that 4, u know that I factored x^2-4...
But why is the limit equal to 4?

Answer 16

The only way to prove it is with epsilon/delta.
But you could also try checking values really close to 2, you'll find it approaches 4.
Try evaluating it at x = 1.9 and x=2.1

Answer 17

I see, you plugged in 2 for x into that equation, YEeeeeeeeees, I think I get it!

Answer 18

Yes, but my point is that \[\Large \lim_{x\rightarrow a}f(x)\]is not necessarily equal to f(a).

Do you get it now?

Answer 19

Yes!

Answer 20

it is limited to a

Answer 21

Good. We have a special name for functions for which it IS true.
IE for functions f such that
\[\Large \lim_{x\rightarrow a}f(x)=f(a)\]

Answer 22

They're called "continuous functions"

Answer 23

Catch me so far?

Answer 24

IE for functions f such that, what is IE?

Answer 25

Latin. Id est.
(meaning: that is to say,)

Answer 26

OK fine!

Answer 27

Okay. Let's have examples of continuous functions.

Answer 28

Actually, sorry, before that, a geometric intuition of continuous functions...

Answer 29

A geometric interpretation of continuity is that you can draw the graph of the function *without lifting your pencil off the paper*

Answer 30

Got that?
this is continuous



this is not.