1. Problem: Calculate the limit $$\lim_{n \to \infty} \frac{n^4 + 5n^2 + n + 2}{3 + 4n - 5n^3 - 10n^4}$$ and similar rational functions of polynomials. Step 1: Identify the highest power of $n$ in numerator and denominator. Step 2: Divide numerator and denominator by the highest power of $n$ in the denominator. Step 3: Simplify the expression and evaluate the limit as $n \to \infty$. Example for the first limit: $$\lim_{n \to \infty} \frac{n^4 + 5n^2 + n + 2}{3 + 4n - 5n^3 - 10n^4} = \lim_{n \to \infty} \frac{1 + \frac{5}{n^2} + \frac{1}{n^3} + \frac{2}{n^4}}{\frac{3}{n^4} + \frac{4}{n^3} - \frac{5}{n} - 10} = \frac{1 + 0 + 0 + 0}{0 + 0 - 0 - 10} = -\frac{1}{10}$$ Repeat similar steps for all given polynomial rational limits. 2. Problem: Calculate limits of the form $$\lim_{x \to 0} \frac{\arctan(a x^m) - \sin(b x^n)}{c x^p}$$. Step 1: Use Taylor expansions near zero: $$\arctan(z) \approx z - \frac{z^3}{3} + \cdots$$ $$\sin(z) \approx z - \frac{z^3}{6} + \cdots$$ Step 2: Substitute $z = a x^m$ and $z = b x^n$. Step 3: Expand numerator and denominator, keep dominant terms. Step 4: Simplify and evaluate the limit. Example: $$\lim_{x \to 0} \frac{\arctan(3x^3) - \sin(7x^4)}{9x^4} = \lim_{x \to 0} \frac{3x^3 - \frac{(3x^3)^3}{3} - (7x^4 - \frac{(7x^4)^3}{6})}{9x^4} = \lim_{x \to 0} \frac{3x^3 - 7x^4 + \text{higher order terms}}{9x^4} = \lim_{x \to 0} \frac{3x^3}{9x^4} - \frac{7x^4}{9x^4} + 0 = 0 - \frac{7}{9} = -\frac{7}{9}$$ 3. Problem: Calculate derivatives of functions like $$f(x) = 8x^2 \cos 2x$$. Step 1: Use product rule: $$(uv)' = u'v + uv'$$ Step 2: Compute derivatives: $$u = 8x^2, u' = 16x$$ $$v = \cos 2x, v' = -2 \sin 2x$$ Step 3: Apply product rule: $$f'(x) = 16x \cos 2x - 16x^2 \sin 2x$$ Repeat for all given functions. 4. Problem: Calculate integrals $$\int f(x) dx$$ for rational functions and trigonometric expressions. Step 1: Use partial fraction decomposition for rational functions. Step 2: Use substitution and standard integrals for trigonometric functions. Step 3: Integrate term by term and simplify. 5. Problem: Determine monotonicity of $$f(x) = 2x^3 + e^{-x^2}$$. Step 1: Compute derivative: $$f'(x) = 6x^2 - 2x e^{-x^2}$$ Step 2: Analyze sign of $f'(x)$ to find intervals of increase/decrease. 6. Problem: Find local extrema using second derivative test for polynomials like $$f(x) = x^5 + 2x^4 + x^2 + 3x^3 + 1$$. Step 1: Compute first derivative $f'(x)$. Step 2: Find critical points by solving $f'(x) = 0$. Step 3: Compute second derivative $f''(x)$. Step 4: Use $f''(x)$ at critical points to determine local minima/maxima. Final answers depend on each specific problem but follow these methods.