calculus - How to evaluate $limlimits_{xto 0} frac{sin x - x + x^3/6}{x^3}$

I'm unsure as to how to evaluate:

$$\lim\limits_{x\to 0} \frac{\sin x - x + \frac{x^3}{6}}{x^3}$$

The $\lim\limits_{x\to 0}$ of both the numerator and denominator equal $0$. Taking the derivative of both ends of the fraction we get:

$$\lim\limits_{x\to 0} \frac{x^2 + 2\cos x -2}{6x^2}$$

But I don't know how to evaluate this?

Many thanks for any help.

Answer

You can use l'Hospital as many times as needed as long as the indeterminate forms conditions are fulfilled. In this case, using Taylor series can be helpful, too:

$$\sin x = x - \frac{x^3}6 + \frac{x^5}{120} - \ldots = x - \frac{x^3}6 + \mathcal O(x^5)$$

$$\implies \frac{\sin x - x + \frac{x^3}6}{x^3} = \frac{\mathcal O(x^5)}{x^3} = \mathcal O(x^2) \xrightarrow[x \to 0]{} 0$$