I will calculate the limits for the given problems one by one. Let's start analyzing them.

Problem 1:

$\lim_{x \to \infty} \frac{2x^4 - 4x + 5}{5x^5 + 3x^4 - 20}$ Here, the highest degree term in the denominator dominates. Let's compute.

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Problem 2:

$\lim_{x \to 8} \frac{x^2 - 64}{\sqrt{x + 1} - 3}$ This is an indeterminate form $\frac{0}{0}$. We'll use algebraic simplification or L'Hôpital's rule.

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Problem 3:

$\lim_{x \to 0} \cot(5x) \cdot \arcsin(7x)$ Since $\cot(5x)$ becomes large as $x \to 0$, but $\arcsin(7x)$ becomes small, we'll analyze their product.

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Problem 4:

$\lim_{x \to \infty} \left(\frac{5x + 7}{5x}\right)^x$ Simplify $\frac{5x + 7}{5x}$ and take the natural logarithm to convert the exponentiation into a product.

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Let me calculate them step by step. sympy import symbols, limit, oo, sqrt, cot, asin

Defining the variable x

x = symbols('x')

Problem 1

expr1 = (2x**4 - 4x + 5) / (5x**5 + 3x**4 - 20) limit1 = limit(expr1, x, oo)

Problem 2

expr2 = (x**2 - 64) / (sqrt(x + 1) - 3) limit2 = limit(expr2, x, 8)

Problem 3

expr3 = cot(5x) * asin(7x) limit3 = limit(expr3, x, 0)

Problem 4

expr4 = ((5x + 7) / (5x))**x limit4 = limit(expr4, x, oo)

(limit1, limit2, limit3, limit4)Here are the computed limits:

1. $\lim_{x \to \infty} \frac{2x^4 - 4x + 5}{5x^5 + 3x^4 - 20} = 0$

2. $\lim_{x \to 8} \frac{x^2 - 64}{\sqrt{x + 1} - 3} = 96$

3. $\lim_{x \to 0} \cot(5x) \cdot \arcsin(7x) = \frac{7}{5}$

4. $\lim_{x \to \infty} \left(\frac{5x + 7}{5x}\right)^x = e^{7/5}$

Would you like detailed steps for any of these solutions?

Related Questions:

1. How do you handle limits involving indeterminate forms like $\frac{0}{0}$?
2. What is the significance of the exponential limit transformation in Problem 4?
3. Can L'Hôpital's rule be used for Problem 2? Why or why not?
4. Why does $\cot(5x) \to \infty$ but the product with $\arcsin(7x)$ remains finite in Problem 3?
5. How do higher-order terms affect limits as $x \to \infty$?

Tip:

When solving limits, always analyze the degree of the numerator and denominator for large $x$ to simplify computations.