Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Okay, so this was labeled as a "fun problem", but I'm having trouble knowing how to approach it.

I'm given: $\lim\limits_{x \to 0^+} f(x) = A$ and $\lim\limits_{x \to 0^-} f(x) = B$.

I need to find (or at least know where to start):

a) $\lim\limits_{x \to 0^+} f(x^3 - x)$

b) $\lim\limits_{x \to 0^-} f(x^3 - x)$

Any insight on how to approach this (or even a solution) would be greatly appreciated.

Thanks

share|improve this question
2  
I guess one of the limits should be $x \to 0^+$, not $0^-$? –  Calle Mar 9 '11 at 23:31
    
Ah yes, my bad. The limit of A is from the right. –  NateyG Mar 9 '11 at 23:50
    
I fixed the limit. You can edit your posts. –  Jonas Meyer Mar 10 '11 at 2:02
add comment

3 Answers

up vote 1 down vote accepted

Let $z(x)=x^3-x$

You have $$\lim_{x\rightarrow 0^- }f(z(x)) =\lim_{z\rightarrow 0^+ }f(z) $$

i.e $z$ approaches from $0+\delta$ when x approaches from $0-\delta$ for $\delta>0$ and vice versa for the other limit.

So whatever limits you have for x in will be inverted in z, i.e, answer for a) is B and b) is A

To get a visual hint, plot $x^3-x$

share|improve this answer
add comment

You know what $f$ does when its input is negative and going to zero and when its input is positive and going to zero. When $x$ is negative and going to zero, then $x^3-x$ is also going to zero, but because $x^2<1$ when $|x|<1$, $x^3-x=x(x^2-1)$ will be positive. Similarly, when $x$ is positive and going to zero, $x^3-x$ will be negative and going to zero. This should allow you to determine the one-sided limits from the information given.

share|improve this answer
add comment

When $x$ goes to zero, $x^3-x$ also goes to zero. But we can be more precise: in fact,

  • when $x$ goes to zero from the left, $x^3-x$ goes to zero from the right, and

  • when $x$ goe to zero from the right, $x^3-x$ goes to zero from the left.

Can you see how to use this to solve the problem?

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.