Landau small o proof

Question

The relationship between two functions f and g is written as $f (x) = o (g (x))$

Show that $tan (x^3 ) = o (x^2 )$

By the epsilon-delta definition, there exist $\epsilon{}>0$ and $\delta{}>0$ such that $|\frac{tan(x^3)}{x^2}|<\epsilon$ whenever $|x-x_0|<\delta{}$. How can you choose $\delta{}$ such that $|\frac{tan(x^3)}{x^2}|<\epsilon$

Answer 1

Exhibiting $\delta$ explicitly in terms of $\varepsilon$ is rather messy. But you can see such $\delta$ exists because the expression is

$$\frac{x}{\cos (x^3)} \frac{\sin(x^3)}{x^3}$$

and we have that

1. $\dfrac{\sin h}{h}\to 1$ as $h\to 0$
2. $\cos h\to 1$ as $h\to 0$, while certainly
3. $h\to 0$ as $h\to 0$.