The correct answer is $\boxed{\text{D}}$.
The limit $\mathop {{\text{Lim}}}\limits_{{\text{x}} \to 0} \frac{{{\text{x}} – \sin {\text{x}}}}{{1 – \cos {\text{x}}}}$ does not exist.
To see this, let’s try to evaluate the limit using direct substitution. We get
$$\mathop {{\text{Lim}}}\limits_{{\text{x}} \to 0} \frac{{{\text{x}} – \sin {\text{x}}}}{{1 – \cos {\text{x}}}} = \frac{{0 – \sin 0}}{{1 – \cos 0}} = \frac{{0 – 0}}{{1 – 1}} = \frac{0}{0}$$
The limit $\frac{0}{0}$ is indeterminate, which means that we cannot evaluate it using direct substitution.
We can try to evaluate the limit using L’Hôpital’s rule. However, L’Hôpital’s rule cannot be applied in this case because the limit $\mathop {{\text{Lim}}}\limits_{{\text{x}} \to 0} \frac{d}{dx} \frac{{{\text{x}} – \sin {\text{x}}}}{{1 – \cos {\text{x}}}}$ is also indeterminate.
Therefore, the limit $\mathop {{\text{Lim}}}\limits_{{\text{x}} \to 0} \frac{{{\text{x}} – \sin {\text{x}}}}{{1 – \cos {\text{x}}}}$ does not exist.