Question

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?

Answer 1

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

Answer 2

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

Answer 3

What's the answer? lol

Answer 4

When in doubt, go with DNE, lol

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

Nice *guess* :\

Answer 10

thnx

Answer 11

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...

Answer 12

*Assume

Answer 13

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

Answer 14

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...

Answer 15

teach me how to do this