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.