$$\mathop {{\text{Lim}}}\limits_{{\text{x}} \to \infty } \left( {\frac{{{\text{x}} + \sin {\text{x}}}}{{\text{x}}}} \right)$$ equal to A. $$ – \infty $$ B. 0 C. 1 D. $$\infty $$

The correct answer is $\boxed{\text{D}}$.

As $x$ approaches infinity, the sine function oscillates between $-1$ and $1$, while the linear function increases without bound. Therefore, the ratio of the sine function to the linear function approaches infinity.

Here is a graph of the function $f(x) = \frac{x + \sin x}{x}$ for $x \ge 0$:

[asy]
unitsize(1 cm);

draw((0,-1.2)–(0,1.2),Arrows);
draw((0,0)–(10,0),Arrows);

real g(real x) {
return x + sin(x);
}

real f(real x) {
return g(x)/x;
}

draw(graph(f,0,10),red);
draw(graph(g,0,10),blue);

label(“$x$”, (10,0), E);
label(“$y$”, (0,1.2), N);
label(“$f(x)$”, (5,1.1), S);
label(“$g(x)$”, (5,-0.1), S);
[/asy]

As you can see, the graph of $f(x)$ approaches infinity as $x$ approaches infinity.