Let's go through each statement one by one to evaluate their correctness.

1. First Statement: $f(x) = \cos(x), \quad g(x) = x^2 - 3. \text{ Then } f(x) \cdot g(x) \text{ is an infinitesimal function as } x \to 0^-.$ Here, as $x \to 0^-$, we have:

   • $\cos(0) = 1$,
   • $g(x) = x^2 - 3$ tends to $-3$.

   So, $f(x) \cdot g(x) = 1 \cdot (-3) = -3$, which is a constant, not infinitesimal. Hence, this statement is false.

2. Second Statement: $f(x) = \sqrt[3]{x} - 1, \quad g(x) = x^3 + 1. \text{ Then } f(x) \cdot g(x) \text{ is an infinitely large function as } x \to 1^-.$ As $x \to 1^-$:

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

   The product tends to $0$, which contradicts the statement about being "infinitely large". Therefore, this statement is false.

3. Third Statement: $\text{The difference } f(x) = \arcsin(x) \text{ and } g(x) = x^2 \text{ as } x \to 0 \text{ is an infinitesimal of higher order compared to both } f(x) \text{ and } g(x).$ As $x \to 0$:

   • $f(x) = \arcsin(x) \sim x$,
   • $g(x) = x^2$.

   The difference $f(x) - g(x) \sim x - x^2$, and since $x^2$ tends to zero faster than $x$, this means $x - x^2$ is of the same order as $f(x)$. Hence, this statement is false.

4. Fourth Statement: $f(x) = \cos(x), \quad g(x) = e^{1/x}. \text{ Then } f(x) \cdot g(x) \text{ is an infinitesimal function as } x \to 0^-.$ As $x \to 0^-$:

   • $f(x) = \cos(0) = 1$,
   • $g(x) = e^{1/x} \to +\infty$.

   The product tends to infinity, not zero. Thus, this statement is false.

5. Fifth Statement: $f(x) = \sqrt[3]{x}, \quad g(x) = \cos(x). \text{ The ratio } \frac{g(x)}{f(x)} \text{ has a finite limit as } x \to 0^+.$ As $x \to 0^+$:

   • $f(x) = \sqrt[3]{x} \to 0$,
   • $g(x) = \cos(0) = 1$.

   The ratio $\frac{g(x)}{f(x)} = \frac{\cos(x)}{\sqrt[3]{x}}$ tends to infinity as $x \to 0^+$. Hence, this statement is false.

6. Sixth Statement: $f(x) = \sqrt[3]{x}, \quad g(x) = \sin(x). \text{ Then } f(x) + g(x) \text{ is an infinitesimal function as } x \to 0^+.$ As $x \to 0^+$:

   • $f(x) = \sqrt[3]{x} \to 0$,
   • $g(x) = \sin(0) = 0$.

   The sum tends to zero, which means it is an infinitesimal function. Hence, this statement is true.

Conclusion:

Only the sixth statement is correct.

Do you have any questions or need further explanation? Here are 5 related questions for deeper exploration:

1. What is the concept of an infinitesimal function?
2. How do you determine the behavior of functions as $x \to 0$?
3. What is the difference between an infinitesimal and infinitely large function?
4. Can you explain the difference between $x \to 0^+$ and $x \to 0^-$?
5. What is the order of infinitesimals, and how does it affect function comparison?

Tip: Always check the limit of a function carefully as $x$ approaches specific values to classify its behavior (infinitesimal, infinite, or finite).