The given expression is: $\lim_{n \to \infty} \frac{\sqrt{1} + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{n}}{8n + 10n^{3/2}}$

Step 1: Analyzing the Sum in the Numerator

The numerator can be expressed as: $\sum_{k=1}^{n} \sqrt{k}$

This sum is approximately equivalent to the integral: $\int_{1}^{n} \sqrt{x} \, dx$ Calculating this integral: $\int_{1}^{n} \sqrt{x} \, dx = \int_{1}^{n} x^{1/2} \, dx = \left[ \frac{2}{3} x^{3/2} \right]_{1}^{n}$ $= \frac{2}{3} \left( n^{3/2} - 1 \right)$

For large values of $n$, the term $n^{3/2}$ dominates, so the numerator is asymptotically equivalent to $\frac{2}{3}n^{3/2}$.

Step 2: Analyzing the Denominator

The denominator is given by: $8n + 10n^{3/2}$

As $n$ becomes very large, the term $10n^{3/2}$ will dominate over $8n$, so the denominator is asymptotically equivalent to $10n^{3/2}$.

Step 3: Simplifying the Limit

Now, we can rewrite the expression as: $\lim_{n \to \infty} \frac{\frac{2}{3}n^{3/2}}{10n^{3/2}}$ $= \lim_{n \to \infty} \frac{\frac{2}{3}}{10} = \frac{2}{30} = \frac{1}{15}$

Conclusion

The value of the given limit is: $\frac{1}{15}$

Would you like more details or have any questions on this solution?

Here are some related questions that may expand on this topic:

1. How do you approximate sums using integrals in calculus?
2. Why does the dominant term in an expression determine the asymptotic behavior?
3. What are some techniques for evaluating limits of sequences involving summation?
4. How can you formally justify the use of asymptotic approximations?
5. What is the difference between evaluating a finite sum and an infinite sum?

Tip: When dealing with limits involving summations, it is often helpful to use integral approximations to understand the behavior of the series as $n$ approaches infinity.