1. We are tasked with finding the limit $$\lim_{x\to +\infty} x \cos(x)$$ 2. The cosine function $\cos(x)$ oscillates between $-1$ and $1$ for all real numbers $x$. 3. The factor $x$ grows without bound as $x \to +\infty$. 4. Because $x$ grows infinitely large but $\cos(x)$ oscillates, the product $x \cos(x)$ oscillates between $-x$ and $x$, which become arbitrarily large in magnitude. 5. Therefore, the limit does not approach a finite number or infinity; it oscillates indefinitely without limit. 6. Hence, $$\lim_{x\to +\infty} x \cos(x) \text{ \textbf{does not exist}}.$$