Question

Can someone explain limits to me really well. I don't get them at all.

Answer 1

The pure mathematical definition of a limit is actually a bit involved. I'm guessing that what you are really looking for is how to determine limits and also have a good general idea of them without the deep esoteric pure mathematical definition.

Answer 2

You get really really really close to a number, but you don't touch it.

Answer 3

Yes, I want to know how to determine them without all the confusing stuff, because it's only making me get them wrong. They're a blurry mess at the moment.

Answer 4

The exposure that most students have at a bachelor's level, especially in applied mathematics as opposed to pure mathematics, revolves around things like rationizing denominators and eliminating values of zero in the denominator. Also, depends on if you are taking a limit of a variable as it goes to a finite value or an infinite value.

Answer 5

If you have a couple examples, we could go from there.

Answer 6

A simple example is \[\lim_{x \rightarrow 0+} 1/x\]

Answer 7

I'm talking about limits of graphs, where x approaches -4^+, where the there are holes in graphs, and where some limits don't exist. One example I have is Lim f(x) and a x approaches 5 on the bottom.

Answer 8

I also forgot to mention that I was confused about whether asymptotes are important to limits.

Answer 9

Ok, we'd have to look at the function to see if it is defined at 5, and perhaps if it is continuous at 5 or whatever value. Asymptotes can help conceptually and visually but are not absolutely necessary for determining limits.

Answer 10

Would you like me to take a picture of the worksheet?

Answer 11

Since there is a graph

Answer 12

Ifthat will help you. But one thing, I don't have Word, but I could look at a jpeg or some other picture. Again, if this will help you.

Answer 13

Hold on.

Answer 14

As you are taking your picture of the graph, are you using L'Hopital's Rule yet or are you at an introduction to limits?

Answer 15

Sounds like a purely visual limit inspection. Looking for continuity, right-side limit equal to left-side limit, etc.

Answer 16

1: http://twitpic.com/b23eow/full
2nd w/ asymptotes: http://twitpic.com/b23etb/full
I'm not sure what L'hospitals rule is...

Answer 17

If it helps, I'm in pre-calc.

Answer 18

Yes, that helps. It also helps that there is not an explicitly-defined function here.

Answer 19

Look at x = -4 for a moment.

Answer 20

I don't think what I need to understand is extremely complex. Just the basics, of looking where x approaches zero and understanding what the "limit" is. Problem is, I always seem to get that wrong.

Answer 21

f(x) = 1, but that is not the limit as x -> -4 from either the right or the left.

Answer 22

What you are essentially looking for is what value of f(x) are you about to get as x approaches -4 "from the left" and ther is a separate limit of f(x) "from the right".

Answer 23

So, look at values of x around -5, going larger (-4.8, -4.5, -4.1, all approximately), just before you get to x = -4.

Answer 24

f(x) is going up in a smooth curve and the limit of f(x) as x approaches -4 "from the left" is 5. With me so far?

Answer 25

LIGHTBULB! For x--> it's not the black dot (1) because it is not approaching 4! Correct?

Answer 26

*-4

Answer 27

It doesn't matter whether f(x) = 5 at x = -4, or even whether or not f(x) is even defined at x = -4. About lightbulb. You're starting to get it, I am sure.

Answer 28

To consolidate your understanding, the limit of f(x) as x approaches -4 "from the RIGHT" is different and is 3.

Answer 29

Sorry, -3.

Answer 30

Again, it doesn't matter that the actual f(-4) is 1. In, short, it's the "open holes".

Answer 31

How would you know if it was approaching from the right or left? Is that why some are x--> 2^+?

Answer 32

Similarly, with the lim of f(x) as x approaches 2+ (or 2 "from the right") is the same as the limit as x approaches 2- (or 2 "from the left") which limit is 5, even though f(x) is not even defined at x = 2.

Answer 33

In notation the direction of the limit is after the number, so the limit of f(x) as x approaches a- means positive a, but from the left, that is, going from left to right.

Answer 34

I see

Answer 35

-> a with no plus or minus means BOTH directions. Sometimes, as with x -> -4, the limit does not exist, because it is different from the left and from the right. So the left and right side limits can exist but not the "combined" limit.

Answer 36

So, -> a is different from -> a+ which is also different from -> a-

Answer 37

So if there are two lines that line up together, the limit does not exist, assuming there is not + or - involved?

Answer 38

I think you meant to say the DON'T line up together , is that right?

Answer 39

Yes, I'm sorry XD

Answer 40

np. Yes, you are definitely getting it now. You just have to see a few more examples and then mull it over a little bit, and it WILL become ingrained in your analytic ability through your subconcious. So, the limit as x -> -4 does not exist but the limits as x -> -4- and as x -> -4+ DO exist. Cool, no?

Answer 41

Yes, cool.

Answer 42

I love it when I work with a good student such as yourself, especially when the lightbulb went on as it did here. But if you were still having troubles, I would work with you as long as you wanted.

Answer 43

Awwwwww. Thank you. I'm definitely starting to understand this more. The last part of my question would be graph 2 and the asymptotes, where i am confused as to whether the answers would be infinity? :)

Answer 44

Let's look at the vertical asymptote at x = 1 first. The limit of f(x) as x -> 1+ is different from the limit of f(x) as x -> 1-. The first limit is infinity (positive infinity) and the second limit is negative infinity. So, no limit on f(x) as x -> 1 (that is, with NO sign after the "1").

Answer 45

Ok, so x-->-2+ would be infinity, but x-->-2- is negative infinity?

Answer 46

Well, you almost got it, but you are still thinking well. The limit of f(x) as x -> -2+ is NEGATIVE infinity (goes DOWN). Good to start getting in the habit of writing or at the very least thinking, that the limit is on f(x) where the notation is saying that "as x approaches some value". Similar for x = -2, the limits go in opposite directions. Now, the limit as x -> infinity (which is the same as + infinity, but it's customary not to put a "+" sign in front of infinity when you want positive infinity) is 1.

Answer 47

Also, one never puts either "+" or "-" AFTER infinity because infinity implies a RAY and one-way direction.

Answer 48

OHHH, So the directions are different! I understand my mistake now.

I cannot thank you enough. This actually concludes my test review. I'm sure I'll do great on my exam tomorrow. Also, you were very patient, for which I am grateful. :)

Answer 49

You are the best. A true pleasure to work with. Have a wonderful night, and good luck ALWAYS.

Answer 50

To you too.

Answer 51

Good stuff. Good stuff. :-)