1. The problem asks us to find $$\lim_{x \to \infty} \cos\left(x^{2} + e^{\frac{x!}{2}}\right)$$. 2. Note that $x!$ (factorial of $x$) grows extremely fast as $x$ increases. Therefore, the exponent $$\frac{x!}{2}$$ also grows very large. 3. Since the exponential function $e^y$ tends to infinity as $y \to \infty$, we have $$e^{\frac{x!}{2}} \to \infty$$ as $x \to \infty$. 4. Consequently, the expression inside the cosine becomes $$x^{2} + e^{\frac{x!}{2}} \to \infty$$. 5. The cosine function is periodic with period $2\pi$ and oscillates between -1 and 1. 6. As the argument inside the cosine tends to infinity without settling to any particular value or pattern, the limit $$\lim_{x\to\infty} \cos\left(x^{2} + e^{\frac{x!}{2}}\right)$$ does not exist because cosine continuously oscillates. **Final answer:** The limit does not exist because the argument tends to infinity and cosine oscillates infinitely.