## Callisto Group Title 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? one year ago one year ago

1. satellite73 Group Title

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 Group Title

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

3. slaaibak Group Title

4. slaaibak Group Title

When in doubt, go with DNE, lol

5. satellite73 Group Title

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

6. satellite73 Group Title

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 Group Title

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

8. satellite73 Group Title

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

9. Callisto Group Title

Nice *guess* :\

10. satellite73 Group Title

thnx

11. Callisto Group Title

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 Group Title

*Assume

13. UnkleRhaukus Group Title

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

14. Callisto Group Title

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!!!! Group Title

teach me how to do this