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. Step 1: Identify the highest power of $n$ in numerator and denominator. Step 2: Divide numerator and denominator by $n^4$ (the highest power). 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 each rational function limit. 4. Problem: Calculate limits of the form $$\lim_{n \to \infty} \left(\frac{an + b}{\sqrt{cn^2 + d}}\right)^n$$. Step 1: Simplify the inside expression by dividing numerator and denominator by $n$. Step 2: Express the limit in the form $$\lim_{n \to \infty} \left(1 + \frac{k}{n}\right)^n = e^k$$. Example: $$\lim_{n \to \infty} \left(\frac{2n + 19}{\sqrt{n^2 - 5}}\right)^n = \lim_{n \to \infty} \left(\frac{2 + \frac{19}{n}}{\sqrt{1 - \frac{5}{n^2}}}\right)^n = \lim_{n \to \infty} \left(2 + \frac{19}{n}\right)^n = e^{19}$$ (after proper expansion and logarithm). 5. Problem: Calculate limits involving factorials like $$\lim_{n \to \infty} \frac{(3n + 1)! + (3n - 1)!}{(3n)! (n^4 - 2)}$$. Step 1: Use Stirling's approximation or factorial properties to simplify. Step 2: Factor out the dominant factorial term. Step 3: Simplify and evaluate the limit. 6. Problem: Calculate limits involving expressions with arctangent and sine functions as $$x \to 0$$. Step 1: Use Taylor expansions for $\arctan(x)$ and $\sin(x)$. Step 2: Substitute expansions and simplify numerator and denominator. Step 3: Evaluate the limit by comparing the lowest order terms. 7. Problem: Calculate limits involving expressions like $$\lim_{x \to 0} \frac{2x + 3x^3}{e^{13x} - \cos(2\sqrt{3}x)}$$. Step 1: Use Taylor expansions for $e^{kx}$ and $\cos(mx)$. Step 2: Substitute expansions and simplify numerator and denominator. Step 3: Evaluate the limit by comparing the lowest order terms. Each problem requires careful algebraic manipulation and application of limit laws. Final answers depend on the specific problem but follow the above methods.