Question

Can someone here help me with limits? We just started Calculus and my teacher didn't explain how to do them, and we don't even have a textbook, so I'm clueless.

Answer 1

Wow, limits are an enormous subject. Do you have a specific example that you would like help with?

Answer 2

Yup! I have a worksheet. Let me just scan it.

Answer 3

Don't laugh if everything's wrong... like I said, she never explained it or gave in-class examples.

Answer 4

I literally have no idea what I'm doing ._.

Answer 5

I won't laugh. Seriously. :D I'm a biologist on the site so the serious math people are in for a treat when they see me tutoring in math.

Answer 6

The point is, I'm the one who they are going to giggle at. :D

Answer 7

http://i158.photobucket.com/albums/t120/brinazarski2/img054.jpg only 1-5

Answer 8

Well, I hope you can help me!

Answer 9

Would you like to start on Problem 1 and work through it?

Answer 10

Yup!

Answer 11

So the first problem gives you a function, G(x). It is testing your knowledge of when a function has a limit. And, crucially, when the function does not have a limit.

The function G(x) has two different kinds of discontinuities. One discontinuity at x = 2 is a point discontinuity and the one at x = 0 is an asymptotic discontinuity. Is that much clear?

Answer 12

Not really... this is the first thing we're doing in Calculus, if you want to call it that, and I have no idea what you just said ^^;

Answer 13

I've never had PreCalc either, if it helps. My Trig class literally just jumped into this the other day, and I missed class today because of the AP exam (though this was do today anyway, hah...))

Answer 14

OK. Cool.

Are you comfortable with the function notation and language? Like when I call the line G(x), you know that it is the specific value of the function for that value x of the input?

Answer 15

Sort of... for the most part, I'd say yes

Answer 16

OK. If I *ever* get away from you or say something you don't understand, please stop me and ask for an explanation. :D

You can have two different kinds of functions: they can be continuous or discontinuous. If you can draw the *entire* line without picking up your pencil off the paper, then it is a continuous function. If you have to pick up your pencil to draw it then it is discontinuous.

The function it shows you has two different places where you'd have to pick up your pencil to draw it, one at x = 0 and one at x = 2.

At x = 0, it does this crazy thing and shoots off the top of the paper toward infinity. And at x = 2, one point is missing from the line. You would have to pick your pencil up to draw it and the lines on each side.

Answer 17

Uh.... I THINK I understood you up to the second paragraph... aren't both lines discontinuous?

Answer 18

I'm imagining a continuous lines takes up the whole graph paper, so you pick up the pencil when there's no room in the graph... yes?

Answer 19

All the lines you see are actually from the same one function, G(x). But the function is discontinuous so it looks like more lines. You have to pick up your pencil in a couple places (at x = 0 and x = 2) to draw it so it *looks* like more than one function.

Answer 20

Ah... I don't see why you'd pick it up at x = 0 or x = 2, though....

Answer 21

If you put your pencil down on the far left hand side of the graph and tried to draw the entire thing without picking your pencil up, you couldn't.

You would have to pick it up at x = 0, because as x gets closer and closer to zero, the value of the function just goes up forever toward infinity. There is no way you can draw infinity on a piece of paper.

Then it comes back down on the other (the positive) side of x = 0.

And at x = 2, the point 2 is missing from the line. It is up above it. You would have to pick your pencil up to draw it.

So the function G(x) is discontinuous. And it is discontinuous when x = 0 and x = 2.

Answer 22

I'm sorry. Looking at the last sentence I should have been clearer. The entire function is discontinuous. It is what it is like at x = 0 and x = 2, where the actual discontinuities are, which makes it that way.

Answer 23

I'm actually kinda' replying slowly, so I didn't get to read that yet.... I'm kinda' everywhere right now ^^;; Sorry

Answer 24

I think I somewhat get it...

Answer 25

OK. Discontinuities are very important in understanding limits.

The point of limits is that the actual value of the function *at* the a point (some value of x) does not matter. The only thing you care about is the value the function approaches as it gets *near* to at that value of x.

That is easier with an example.

So let's look at part A of that problem. It asks you, what is the limit of G(x) as x gets close and closer to 2. And beside the 2 you can see a negative sign. That tells you that it wants to approach x from the left.

So if you put your pencil on the line and move it closer and closer toward x = 2 from the left, the function G(x) gets closer and closer to 1.

It does not matter what G(x) actually equals at one. (It happens to equal 3.) All that matters is that x gets closer and closer to 2, the value of G(x) gets closer and closer to 1.

Take your time with that. There is a lot there.

Answer 26

I'm assuming, since I start from the elft, I start with the left line... but then that gets closer to zero...

Answer 27

No, it is asking you about the point x = 2. Not the left of the entire function.

It wants you to put the pencil on G(x) and move it along G(x) toward where x = 2. From the left. It asks you for the value that G(x) gets closer and closer to as you do this.

Answer 28

So... the black dot or white dot?

Answer 29

The black dot is the value that the function G(x) actually has at x = 2. The value is 3.

But for the purposes of finding the limit, we don't need to care what that value is *at* x = 2. We only care about the value of the function as x gets *near* to 2. It looks like it gets closer and closer to 1, so the limit of G(x) from the left is 1.

Answer 30

NOW I know where you got the three from! Is it one because the line is closest to that dot at x = 1?

Answer 31

Nope. Sorry. No.

Answer 32

When you are finding the limit at x = 2, you are asking, "What value does the function G(x) get near to, as x gets close to 2?"

So as you drag your pencil along the line closer and closer to x = 2, the value of the line gets closer and closer to 1.

Answer 33

It looks more like it's getting closer to zero, though...

Answer 34

I am going to try to draw this because a picture is clearer than words.



That is a crude copy of what we are looking at.