The uploaded file contains a series of mathematical problems. Here's the transcription of the text and an overview of the tasks:

Problems:

1. Пусть существует $f'(x)$ на $(0, +\infty)$, существует $\lim_{x \to +\infty} f'(x)$, $f(x) = o(x)$ при $x \to +\infty$. Верно ли, что $\lim_{x \to +\infty} f'(x) = 0$?

2. Доказать, что
   $f(x) = \frac{d^{80}}{dx^{80}}(e^{-x^{80}})$
   имеет 80 положительных корней.

3. Исследовать на сходимость последовательность
   $\{x_n\}, \, x_n = \arctan x_{n-1}, \, x_0 = \frac{1}{2}.$

4. Найти разложение в ряд Тейлора:
   $(1 + x)^{-1} - 1,$
   в точке $x_0 = 0$, до члена с $x^4$.

5. Найти предел:
   $\lim_{x \to +\infty} \frac{\ln(1 + \sin^2 2x)}{\ln(1 + \sin^2 4x)}.$

Let me know which specific problem you'd like to solve first, or I can work on them one by one. Would you like a detailed solution for all of them?

To deepen understanding, here are some exploratory questions:

1. How does the condition $f(x) = o(x)$ affect the behavior of $\lim_{x \to +\infty} f'(x)$?
2. What are the general properties of derivatives of exponential functions, especially $e^{-x^{80}}$?
3. Why does the arctangent sequence converge, and how is the fixed point determined mathematically?
4. What are the rules for expanding rational functions like $(1 + x)^{-1}$ in a Taylor series?
5. How do trigonometric oscillations influence the evaluation of limits with logarithms?

Tip:

For Taylor series, ensure you calculate derivatives systematically and evaluate them precisely at the expansion point.