# Need Help with My Limit Homework

1. Sep 22, 2009

### ModalNekat

1. The problem statement, all variables and given/known data
Limit x approached zero from right which is X * [|1/X|]

2. Relevant equations
Since i Don't know how to draw the graphs, then i don't have an equations

3. The attempt at a solution
I've tried to drew the graphic, but from the graphs, i've concluded that lim x aprroached zero from right [|1/x|] does not exist because 1/0 = undefined.

Actually, the answer is 1, but still i don't know how to solve this.

2. Sep 22, 2009

### VietDao29

$$\lim_{x \rightarrow 0 ^ {+}} \left( x \left| \frac{1}{x} \right| \right)$$?

Well, the first thing when dealing with absolute value, is to take out all absolute signs.

You know that:

$$|a| = \left\{ \begin{array}{ll} a & \mbox{, if } a \geq 0 \\ -a & \mbox{, if } a < 0 \end{array} \right.$$

So, when x tends to 0+, (i.e it tends to 0 from the right), is 1/x positive or negative? Can you break absolute signs?

After breaking (taking out) all the absolute signs, you should be arriving to the final answer shortly. :)

3. Sep 23, 2009

### ModalNekat

Whoops, i'm sorry for misunderstanding, looks like [| |] is not dealing with Greatest INteger Function. Then, what i mean with [|x|] is [[ x ]], sorry..

4. Sep 23, 2009

### VietDao29

You mean the http://en.wikipedia.org/wiki/Floor_and_ceiling_functions" [Broken], right? Or are you refering to the Floor Function?

$$\lim_{x \rightarrow 0 ^ {+}} \left( x \left\lceil \frac{1}{x} \right\rceil \right)$$

If you really mean the Ceiling Function, then from its definition, we can derive to the following inequality:

$$\frac{1}{x} \leq \left\lceil \frac{1}{x} \right\rceil \leq \frac{1}{x} + 1$$

Can you find the limit of the expression based on this inequality?

Last edited by a moderator: May 4, 2017
5. Sep 23, 2009

### Bohrok

I've always known [|x|] to be the greatest integer function, or floor function.
Are you saying it actually looks like [[x]] instead?

6. Sep 24, 2009

### ModalNekat

Yeah, i mean it's the ceiling function. Thanks a lot, right now it could be derived.Actually i got this problems from Purcell's Calculus Book, do u have one on you??

Last edited by a moderator: May 4, 2017
7. Sep 26, 2009

### emyt

8. Sep 26, 2009

### slider142

Use the Squeeze Theorem on the inequality in post #4.

9. Sep 26, 2009

### emyt

the interval between 1/x and 1/x + 1 is fairly large though?

10. Sep 26, 2009

### slider142

Multiply through by x, then take the limit of each expression. Remember that x is not zero; the limit only examines the behavior of these expressions in arbitrarily small neighborhoods of 0. Use the Squeeze Theorem to imply the limit of the central expression.

11. Sep 26, 2009

### emyt

right, thanks