Find Function Satisfying Conditions: x→0

[katkat]

Does anyone know a function t(x) satisfying the following conditions.

as x -> 0 we have

(1) t(x)-> oo

(2) x t(x) -> 0

(3) x t(x) exp(t(x)) -> 0

(4) x t(x)^2 exp(t(x)) -> oo

I have "two extremes" as

t(x) = sqrt(-log x) satisfies (1-3) but x t(x)^2 exp(t(x)) -> 0

t(x) = -log( -x log(x))

satisfies (1-2) and (4), but x t(x) exp(t(x)) -> c >0

I have been searching desperately.

 

[whozum]

What level course is this?

 

[steven187]

hello

if your really desperate you should try ploting it through mathematica or some similar program and watch what happens as x goes to infinity it shouldn't be too hard, a few guesses and you should be able to get your solution

 

[rachmaninoff]

This is in the reals, right? And am I misreading this:

as x -> 0 we have

(1) t(x)-> oo

(2) x t(x) -> 0

or are (1) and (2) mutually exclusive?

 

[whozum]

This is in the reals, right? And am I misreading this:


or are (1) and (2) mutually exclusive?

I want to agree with you but first I want to verify that the only functions that tend to infinity as x-> 0 are ones where the denominator has a quicker tendency to infinity than the numerator.

 

[joeboo]

This is in the reals, right? And am I misreading this:


or are (1) and (2) mutually exclusive?

$$t(x) = \frac{1}{\sqrt{\vert x \vert}}$$

Am I missing something?

 

[whozum]

$$\lim_{x\rightarrow 0} \frac{x}{\sqrt{|x|}} \neq 0$$

 

[rachmaninoff]

edit: I misread the OP, ignore this post.

 

[shmoe]

$$\lim_{x\rightarrow 0} \frac{x}{\sqrt{|x|}} \neq 0$$

It's zero. When x>0 it's sqrt(x). It appears he's only looking at right handed limts (hence his log's), but for this function the two-sided limit is also zero, it's -sqrt(x) for x<0.

These limits are as x->0, none of them exclude any others.


t(x) = -log( -x log(x)) is hopeful. In condition 3, you're looking at [-log(x)-log(-log(x))]/(-log(x)). You'd like to speed up the growth of the denominator without affecting the numerator signifigantly. Try fiddling with the inside log of t(x)

 

[whozum]

I thought of this last night, does it have to be continuous?
And also, for a limit to exist, the function must approach the value from both sides, however since the log function can't take negative values, there is a whole in his domain for t(x) = -log( -x log(x)), correct?

 

[shmoe]

The log(x) is what made me think he was considering right hand limits. Finding a function satisfying those limits from the right is enough in any case, just put |x| where you see x and you've got your two sided limits as well. There's nothing about continuity as he's written it, just a limit. In fact the first condition rules on continuity at 0. The t(x) = -log( -x log(x)) will work, with some minor alterations.