1. Stating the problem: Evaluate the limit $\lim_{n \to \infty} n \sqrt{5n + 3}$. 2. Rewrite the expression inside the limit: $$n \sqrt{5n + 3} = n (5n + 3)^{1/2}$$ 3. Factor $n$ inside the square root to express in terms of $n$: $$(5n + 3)^{1/2} = \left(n \left(5 + \frac{3}{n}\right)\right)^{1/2} = n^{1/2} \left(5 + \frac{3}{n}\right)^{1/2}$$ 4. Substitute back: $$n (5n + 3)^{1/2} = n \cdot n^{1/2} \left(5 + \frac{3}{n}\right)^{1/2} = n^{3/2} \left(5 + \frac{3}{n}\right)^{1/2}$$ 5. Analyze the behavior as $n \to \infty$: - $\left(5 + \frac{3}{n}\right)^{1/2} \to \sqrt{5}$ because $\frac{3}{n} \to 0$. - $n^{3/2} \to \infty$ because the power is positive. 6. Therefore, $$\lim_{n \to \infty} n \sqrt{5n + 3} = \lim_{n \to \infty} n^{3/2} \sqrt{5 + \frac{3}{n}} = \lim_{n \to \infty} n^{3/2} \times \sqrt{5} = \infty$$ 7. Conclusion: The limit diverges to infinity. Final answer: $\boxed{\lim_{n \to \infty} n \sqrt{5n + 3} = \infty}$.