Let's solve these limits one by one by analyzing the behavior as $n \to \infty$. Each expression involves a rational function, so we’ll simplify each limit by identifying the highest power of $n$ in both the numerator and the denominator.

Problem (1)

$\lim_{n \to \infty} \frac{2n^3 - 7n^2 + 8n^5}{5n^5 + 4n^2 - 2}$ The highest power of $n$ in both the numerator and denominator is $n^5$. Dividing every term by $n^5$, we get: $\lim_{n \to \infty} \frac{\frac{2n^3}{n^5} - \frac{7n^2}{n^5} + \frac{8n^5}{n^5}}{\frac{5n^5}{n^5} + \frac{4n^2}{n^5} - \frac{2}{n^5}} = \lim_{n \to \infty} \frac{0 - 0 + 8}{5 + 0 - 0} = \frac{8}{5}.$

Problem (2)

$\lim_{n \to \infty} \frac{2n^2 - 7}{\sqrt{5n^4 + 14n^2}}$ The denominator can be rewritten as $\sqrt{5n^4 + 14n^2} = n^2 \sqrt{5 + \frac{14}{n^2}}$. Dividing the numerator and denominator by $n^2$, we obtain: $\lim_{n \to \infty} \frac{\frac{2n^2}{n^2} - \frac{7}{n^2}}{\sqrt{5 + \frac{14}{n^2}}} = \frac{2 - 0}{\sqrt{5 + 0}} = \frac{2}{\sqrt{5}}.$

Problem (3)

$\lim_{n \to \infty} \frac{2n^2 - 7n}{-5n + 14}$ Dividing both the numerator and the denominator by $n$, we get: $\lim_{n \to \infty} \frac{\frac{2n^2}{n} - \frac{7n}{n}}{\frac{-5n}{n} + \frac{14}{n}} = \lim_{n \to \infty} \frac{2n - 7}{-5 + 0} = -\infty.$ Since the term $2n$ grows without bound, this limit diverges to $-\infty$.

Problem (4)

$\lim_{n \to \infty} \frac{2n^2 + 4n}{\sqrt{5n^2 + 14}}$ We can factor out $n$ in the denominator: $\sqrt{5n^2 + 14} = n \sqrt{5 + \frac{14}{n^2}}.$ Dividing the numerator and the denominator by $n$, we have: $\lim_{n \to \infty} \frac{\frac{2n^2}{n} + \frac{4n}{n}}{n \sqrt{5 + \frac{14}{n^2}}} = \frac{2 + 0}{\sqrt{5 + 0}} = \frac{2}{\sqrt{5}}.$

Summary of Results

1. $\lim_{n \to \infty} \frac{2n^3 - 7n^2 + 8n^5}{5n^5 + 4n^2 - 2} = \frac{8}{5}$
2. $\lim_{n \to \infty} \frac{2n^2 - 7}{\sqrt{5n^4 + 14n^2}} = \frac{2}{\sqrt{5}}$
3. $\lim_{n \to \infty} \frac{2n^2 - 7n}{-5n + 14} = -\infty$
4. $\lim_{n \to \infty} \frac{2n^2 + 4n}{\sqrt{5n^2 + 14}} = \frac{2}{\sqrt{5}}$

Would you like a detailed explanation of any specific part of these calculations?

Here are some related questions:

1. How can you determine the dominant term in a polynomial limit?
2. What happens to rational functions as $n \to \infty$ when the numerator's degree is higher than the denominator’s?
3. How do square roots affect the growth rate of terms involving polynomials?
4. What is the impact of a negative coefficient in the denominator on the limit?
5. Why does dividing each term by the highest power of $n$ simplify limits as $n \to \infty$?

Tip: When dealing with limits of sequences or functions at infinity, always compare the highest power terms in the numerator and denominator to determine the dominant behavior.