1. **State the problem:** Find the limit $$\lim_{x \to +\infty} 3^{\left(\frac{1}{2}\right)^x} 4^x + 2$$. 2. **Recall the properties and formulas:** - As $x \to +\infty$, $\left(\frac{1}{2}\right)^x \to 0$ because $0 < \frac{1}{2} < 1$. - Exponential functions with base greater than 1 grow without bound as $x$ increases. 3. **Analyze the expression:** - The term $3^{\left(\frac{1}{2}\right)^x}$ approaches $3^0 = 1$ as $x \to +\infty$. - The term $4^x$ grows exponentially to $+\infty$ as $x \to +\infty$. 4. **Combine the terms:** - Since $3^{\left(\frac{1}{2}\right)^x} \to 1$, the product $3^{\left(\frac{1}{2}\right)^x} 4^x \approx 1 \cdot 4^x = 4^x$ for large $x$. - Adding 2 does not affect the limit's behavior at infinity. 5. **Conclusion:** - Therefore, $$\lim_{x \to +\infty} 3^{\left(\frac{1}{2}\right)^x} 4^x + 2 = +\infty$$ because $4^x$ dominates and grows without bound. **Final answer:** $$\boxed{+\infty}$$