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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.


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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. – Nate222 Mar 9 '11 at 23:50
I fixed the limit. You can edit your posts. – Jonas Meyer Mar 10 '11 at 2:02
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 $x^3-x$

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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?

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