# Show that $\lim_{x\to0}f(x) = \lim_{x\to0}f(x^3)$ (Real Analysis)

I'm trying to prove that $$\lim_{x\to0}f(x) = \lim_{x\to0}f(x^3)$$ (The domain is not specified and neither the continuity, so it really is only about the limit of an arbitrary function)..

I'm guessing I need to simply use the definition. However I'm not sure where to start.

Here is what I had in mind, however I feel like it's not at all how it should be proved.

Proof

Let $$\lim_{x\to0}f(x) = L$$.

Then, we have that

$$\forall \epsilon > 0, \exists \delta>0 \text{ such that whenever } 0<|x|< \delta \Rightarrow |f(x) - L| < \epsilon$$

Let $$y = x^3$$, then we want to show that $$|f(y) - L| < \epsilon \text{ whenever } 0<|x| = |y^{\frac{1}{3}}| < \delta$$

And here is where I get stuck. Is there any more efficient way to prove this?

Thank you!

Asserting that $$\lim_{x\to0}f(x)=L$$ means, as you wrote, that$$(\forall\varepsilon>0)(\exists\delta>0):\lvert x\rvert<\delta\implies\bigl\lvert f(x)-L\bigr\rvert<\varepsilon.$$So, take $$\delta^\star=\sqrt[3]\delta$$ and then$$\lvert x\rvert<\delta^\star\implies\lvert x^3\rvert<\delta\implies\bigl\lvert f(x^3)-L\bigr\rvert<\varepsilon.$$In other words, $$\lim_{x\to0}f(x^3)=L$$. Can you do it in the opposite direction now?

As an alternative we can proceed by composite function, assuming that $$\lim_{x\to0}f(x)=L$$ it means that

$$\forall\epsilon>0\quad \exists\delta>0\quad |x|<\delta\quad \bigl\lvert f(x)-L\bigr\rvert<\epsilon$$

Now consider $$g(x)=x^3$$ and we have that $$\lim_{x\to0}g(x)=0$$ that is

$$\forall\epsilon_1>0\quad \exists\delta_1>0\quad |x|<\delta_1\quad\bigl\lvert x^3\bigr\rvert<\epsilon_1$$

then if we assume $$\delta=\epsilon_1$$ we have that

$$\forall\epsilon>0\quad \exists\delta_1>0\quad |x|<\delta_1\quad \bigl\lvert f(g(x))-L\bigr\rvert<\epsilon$$

that is

$$\lim_{x\to0}f(g(x))=\lim_{x\to0}f(x^3)=L$$

and similarly for the opposite direction.