Question

(calculus) "Suppose the function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of lim x->0 f(x)? Give reasons for your answer."
There's no graph, so I'm not really sure how to figure this out.

Answer 1

If f(x) is not continuous, the answer is no.

Answer 2

f(x) =1 if x<0
f(x)=2 if x >=0

Answer 3

I'm not sure how you came to this conclusion. It makes sense to me if there were an equation or graph, but it's the abstract theory that's throwing me. How do you know that f(x)=1 if x<0?

Answer 4

you have to construct a counter example to show that the limit does not exist. I constructed one.

Answer 5

But how do you know the limit does not exist? There's no equation or graph to work with.

Answer 6

The right hand limit of f at x=0 is 2 and
The left hand limit of f at x=0 is 1
So the limit at x=2 does not exist.

Answer 7

How did you come up with that though? There's no equation or graph. Where are you getting these limits? I mean I get how a limit doesn't exist if the limit from the right is different from the limit from the left, that I get. I just don't understand where you're getting an equation when I can't see one.
I looked at the solution in the book and it says "nothing can be said."

Answer 8

I think you should read more about limits. Sorry, you should that before we can continue this discussion.

Answer 9

you should do that

Answer 10

hm. I guess the 1.5 hours a day, Monday through Friday classes of Calculus, and the 6 total weekend hours I've been reading the book and the 8 pages of notes from lecture aren't enough. I'll email my professor and SI instructor, that'll probably make it clearer.

Answer 11

Do that.

Answer 12

In the mean time, see the attached graph of the example, I gave.

Answer 13

You can see that the function has a jump at x=0 from 1 to 2. So it does not a limit.

Answer 14

does not have a limit.

Answer 15

okay, I think you aren't understanding /where/ I'm getting stuck. I get that if a limit from the right is say 2 and a limit from the left is -2, then the limit does not exist.
What I'm wondering is why you're giving me equations when the original problem doesn't have one. It's asking if anything can be said about \[\lim_{x \rightarrow 0} f(x)\] It's not asking for examples, but the reasoning. Words, not equations.

Answer 16

just putting in my two cents, I could be so wrong, but it says that is it defined for all x in [-1,1], we can't assume where it is defined

Answer 17

eliassaab is talking about strategies that are being used in proofs, we give counterexamples all the time in the higher mathematics.

Answer 18

All you have to do is to provide an example to say that if a function is defined everywhere. It does not have a limit in general at every point.

Answer 19

this looks like an open ended question where your reasoning is being challenged and they are trying to determine if you understand the defintion of a limit.

Answer 20

Ah gotcha @precal, sooper helpful. The last Calculus I took was for the life sciences, I changed my degree and am now taking the proper Calc. The manual and my SI leader aren't asking for examples, but the reasoning.

Answer 21

open ended questions can be answered in many ways, there is no one correct response but if you give a response you do have to watch your logic because if there is a flaw your professor will find it

Answer 22

you can use specific examples to help you write general examples
general examples will cover all cases and is more appropriate in the upper level math courses

Answer 23

Oh okay. So the book is saying "nothing can be said" about \[\lim_{x \rightarrow 0}f(x)\]
None of the examples show equations, and the SI leader said she wanted written out explanations, like reasoning. So... hrm. Can nothing be said because there isn't enough information in the question to draw succinct conclusions?

Answer 24

well earlier you used the definition of a limit,
example
in order for a limit to exist, the right hand limit and left hand limit have to be the same value
it is not uncommon to maybe given various examples of you now knowing that both sides have the same limit when x approaches 0 from the left side and the right side
you are on the correct path (just second guessing yourself)
let me find you the best video I think I have found on limits

Answer 25

http://www.youtube.com/watch?v=EX_is9LzFSY
he talks fast but after you watch him you will get a better idea of the graphs of limits
watch his examples of graphs of limits

Answer 26

since you have a basic background in calculus, you will enjoy this video
good luck

Answer 27

Thank you @precal, you've been a phenomenal help here.

Answer 28

yw
this is a great site to just come and talk about what you are studying or just to go over concepts
limits can be tricky for the beginner
hang in there