Let's analyze the two given statements:

Statement (i):

If $\lim_{x \to \infty} f(x) = \infty, \quad \lim_{x \to \infty} g(x) = \infty, \quad \text{and} \quad \lim_{x \to \infty} \frac{f(x)}{g(x)} = \infty,$ then $f(x) \to \infty$ slower than $g(x) \to \infty$.

Explanation:
The condition $\lim_{x \to \infty} \frac{f(x)}{g(x)} = \infty$ means that $f(x)$ grows much faster than $g(x)$ as $x \to \infty$. Therefore, the statement that $f(x)$ grows slower than $g(x)$ is incorrect.
This statement is False.

Statement (ii):

If $\lim_{x \to \infty} f(x) = \infty, \quad \lim_{x \to \infty} g(x) = \infty, \quad \text{and} \quad \lim_{x \to \infty} \frac{f(x)}{g(x)} = 0,$ then $f(x) \to \infty$ faster than $g(x) \to \infty$.

Explanation:
Here, $\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0$ indicates that $f(x)$ grows slower than $g(x)$. So, the conclusion that $f(x)$ grows faster is incorrect.
This statement is False.

Final Answer:

• (i) is False.
• (ii) is False.

Would you like further clarification or more details on this? Here are 5 related questions that may help expand your understanding:

1. What does it mean when a limit of a ratio of two functions tends to infinity?
2. How can we compare the growth rates of two functions as $x$ approaches infinity?
3. What does L'Hopital's rule say about evaluating limits of indeterminate forms?
4. What are common examples of functions with different growth rates (e.g., exponential, polynomial)?
5. Can a function grow faster than another even if both approach infinity?

Tip: When comparing functions' growth rates, always check the limit of their ratio to see which one dominates as $x \to \infty$.