# Evaluate $\lim_{x\to 0} \frac{f(x^3)}{x}$

Let $$f:\mathbb{R}\to \mathbb{R}$$ be a function such that $$|f(x)|\leq 2|x|$$ for every $$x \in \mathbb{R}$$. Evaluate $$\lim_{x\to 0} \frac{f(x^3)}{x}$$.

According to the answer key, it is $$0$$ (which matches mine). I am not so sure about my solution (below) though.

Rewritting $$\lim_{x\to 0} \frac{f(x^3)}{x}$$ yields

$$\lim_{x\to 0} \frac{f(x^3)}{x} = \lim_{x\to 0} 2\frac{|x^3|}{x} = 2\lim_{x\to 0} \frac{|x^3|}{x}$$

Analyzing one-sided limits, we have:

$$\lim_{x\to 0^-} \frac{-x^3}{x}=\lim_{x\to 0^-} -x^2=0$$ and $$\lim_{x\to 0^+} \frac{x^3}{x}=\lim_{x\to 0^+} x^2=0$$

Both one-sided limits exist and are equal, therefore $$2\lim_{x\to 0} \frac{|x^3|}{x}= 2\cdot0=0$$.

Is my solution correct?

Subtle points: you should have $$0 \leq \left|\lim_{x\to 0}\frac{f(x^3)}{x}\right|\leq \left|\lim_{x\to 0}\frac{2|x^3|}{x}\right|;$$then you can proceed.
• That's how I should've started, right? Then, $0 \leq \left|\lim_{x\to 0}\frac{f(x^3)}{x}\right|\leq \left|\lim_{x\to 0}\frac{2|x^3|}{x}\right|$ $\rightarrow$ $0 \leq \left|\lim_{x\to 0}\frac{f(x^3)}{x}\right|\leq \left|0|$ $\rightarrow$ $\left|\lim_{x\to 0}\frac{f(x^3)}{x}\right| = 0$. Is that correct? Jul 11, 2020 at 0:41