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Let's analyze the statements one by one to determine their correctness. The question is asking for the true (верные) statements.

### First Statement:
\[
f(x) = \sqrt[3]{x}, \quad g(x) = \cos(x).
\]
We are asked whether the limit of the function \(\frac{g(x)}{f(x)}\) has a finite value as \(x \to 0^+\).

- As \(x \to 0^+\):
  - \(f(x) = \sqrt[3]{x} \to 0\),
  - \(g(x) = \cos(x) \to 1\).

So the limit becomes:
\[
\lim_{x \to 0^+} \frac{\cos(x)}{\sqrt[3]{x}} = \lim_{x \to 0^+} \frac{1}{\sqrt[3]{x}} = \infty.
\]
Thus, the limit is infinite, **not finite**.

**This statement is false.**

### Second Statement:
\[
f(x) = \cos(x), \quad g(x) = x^2 - 3.
\]
We are asked whether \(f(x) \cdot g(x)\) is an infinitesimally small function as \(x \to 0^-\).

- As \(x \to 0^-\):
  - \(f(x) = \cos(0) = 1\),
  - \(g(x) = x^2 - 3 \to -3\).

Thus:
\[
f(x) \cdot g(x) = 1 \cdot (-3) = -3.
\]
This is not an infinitesimally small value; it's a constant.

**This statement is false.**

### Third Statement:
\[
f(x) = \sqrt[3]{x}, \quad g(x) = \sin(x).
\]
We are asked whether \(f(x) + g(x)\) is an infinitesimally small function as \(x \to 0^+\).

- As \(x \to 0^+\):
  - \(f(x) = \sqrt[3]{x} \to 0\),
  - \(g(x) = \sin(x) \to 0\).

Thus:
\[
f(x) + g(x) = \sqrt[3]{x} + \sin(x) \to 0.
\]
Both functions tend to zero, so their sum also tends to zero.

**This statement is true.**

### Fourth Statement:
\[
f(x) = \sqrt[3]{x - 1}, \quad g(x) = x^3 + 1.
\]
We are asked whether \(f(x) \cdot g(x)\) is an infinitely large function as \(x \to 1^-\).

- As \(x \to 1^-\):
  - \(f(x) = \sqrt[3]{1 - 1} = 0\),
  - \(g(x) = 1^3 + 1 = 2\).

Thus:
\[
f(x) \cdot g(x) = 0 \cdot 2 = 0.
\]
The product is zero, not infinitely large.

**This statement is false.**

---

### Summary:
- **True Statement**: The third one, about the sum of \(f(x)\) and \(g(x)\) being infinitesimally small as \(x \to 0^+\).

Do you want further details on any specific statement? 

Here are some related questions:
1. What does it mean for a function to be infinitesimally small?
2. How can we calculate limits for composite functions?
3. What happens to trigonometric functions like \(\cos(x)\) and \(\sin(x)\) near zero?
4. How do cube root functions behave around zero?
5. What methods can be used to determine whether a limit is infinite?

**Tip**: Always check the behavior of each individual function at the limit point before combining them in sums or products.

Выберите все верные утверждения из следующих выражений, касающихся пределов и бесконечно малых функций при определенных значениях переменной x.

Explore the behavior of functions and their limits, including cube roots, trigonometric functions, and infinitesimal properties. This problem set challenges university-level calculus students to identify correct statements regarding limits and function behavior as x approaches critical points.

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