The given mathematical expression is:

$\lim_{x \to 0} \frac{\tan x - x}{x - \sin x}.$

To solve this limit, we proceed step-by-step:

Step 1: Expand $\tan x$ and $\sin x$ using Taylor series.

For small $x$, the Taylor expansions are: $\tan x = x + \frac{x^3}{3} + O(x^5),$ $\sin x = x - \frac{x^3}{6} + O(x^5).$

Substitute these into the numerator and denominator:

Numerator: $\tan x - x = \left(x + \frac{x^3}{3}\right) - x = \frac{x^3}{3} + O(x^5).$

Denominator: [