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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.

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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 \(x \to \infty\).

Consider the following statements:
(i) If lim (x → ∞) f(x) = ∞, lim (x → ∞) g(x) = ∞, and lim (x → ∞) (f(x) / g(x)) = ∞, then f(x) → ∞ slower than g(x) → ∞.
(ii) If lim (x → ∞) f(x) = ∞, lim (x → ∞) g(x) = ∞, and lim (x → ∞) (f(x) / g(x)) = 0, then f(x) → ∞ faster than g(x) → ∞.
Determine which of the above statements are True (1) or False (2).

This problem explores the growth rates of functions using limits and compares how f(x) and g(x) approach infinity. Learn how to analyze the asymptotic behavior of functions based on their limit ratios. Suitable for calculus students.

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