## Callisto 2 years ago If $$\lim_{x \rightarrow 0^{+}} f(x)=A$$ and $$\lim_{x \rightarrow 0^{-}} f(x)=B$$, find $$\lim_{x \rightarrow 0^{+}} f(x^3-x)$$ How to start?

1. satellite73

hmmm maybe find out whether $$x^3-x$$ is approaching 0 from the right or the left as x approaches 0 from the right

2. Callisto

How....? FYI, I have the answer :\

3. slaaibak

4. slaaibak

When in doubt, go with DNE, lol

5. satellite73

you can reason as follows: for small positive values of $$x$$ we have $$x^3<x$$ and so $$x^3-x<0$$

6. satellite73

this is my best guess at any rate i am trying to think up a counter example, one where you wouldn't know the limit, but off the top of my head i cannot, so perhaps what i wrote is correct

7. satellite73

in english, as $$x\to 0^+$$ we have $$x^3-x\to 0^-$$

8. satellite73

so my guess is $$B$$ although i have a 50% chance of being right even if my reasoning is faulty

9. Callisto

Nice *guess* :\

10. satellite73

thnx

11. Callisto

Assuming your way to do this question is correct. Similarly, for the question (in part b) $$\lim_{x \rightarrow 0^{-}} f(x^3-x)$$ $x^3-x>0$So, as $$x \rightarrow 0^+$$, $$x^3-x \rightarrow 0^{+}$$. And it is A. Hmmm...

12. Callisto

*Assume

13. UnkleRhaukus

is the limit of the function of a sum , equal to the sum of the limits of the function ?

14. Callisto

As for part c, $\lim_{x \rightarrow 0^{+}} f(x^2-x^4)$ $x^2-x^4>0$ As $$x \rightarrow 0^{+}$$, $$x^2-x^4\rightarrow 0^{+}$$, so it is A. Seems this trick works, but I don't know why...

15. HELP!!!!

teach me how to do this