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Let's analyze and solve each of the limits given the conditions \( \lim_{{x \to a}} f(x) = l \) and \( \lim_{{x \to a}} g(x) = m \).

### Part (a)
\[ \lim_{{x \to a}} f(x)g(x) \]

Using the product rule for limits, we get:
\[
\lim_{{x \to a}} f(x)g(x) = \lim_{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x) = l \cdot m
\]

### Part (b)
\[ \lim_{{x \to a}} \left[ \frac{f(x) + 2g(x)}{2f(x) - g(x)} \right] \quad (\text{when } l, m > 0) \]

Using the sum and quotient rules for limits:
\[
\lim_{{x \to a}} \left[ \frac{f(x) + 2g(x)}{2f(x) - g(x)} \right] = \frac{\lim_{{x \to a}} f(x) + 2\lim_{{x \to a}} g(x)}{2\lim_{{x \to a}} f(x) - \lim_{{x \to a}} g(x)}
\]
Substitute the limits:
\[
\frac{l + 2m}{2l - m}
\]

### Part (c)
\[ \lim_{{x \to a}} \left[ 2f(x) + 3g(x) \right] \]

Using the sum rule for limits:
\[
\lim_{{x \to a}} \left[ 2f(x) + 3g(x) \right] = 2\lim_{{x \to a}} f(x) + 3\lim_{{x \to a}} g(x)
\]
Substituting the limits:
\[
2l + 3m
\]

### Part (d)
\[ \lim_{{x \to a}} \left[ \frac{f(x) + g(x)}{(g(x))^2} \right] \quad (\text{when } m > 0) \]

Using the sum and quotient rules for limits:
\[
\lim_{{x \to a}} \left[ \frac{f(x) + g(x)}{(g(x))^2} \right] = \frac{\lim_{{x \to a}} f(x) + \lim_{{x \to a}} g(x)}{\left(\lim_{{x \to a}} g(x)\right)^2}
\]
Substituting the limits:
\[
\frac{l + m}{m^2}
\]

### Summary of Results
- (a) \( \lim_{{x \to a}} f(x)g(x) = lm \)
- (b) \( \lim_{{x \to a}} \left[ \frac{f(x) + 2g(x)}{2f(x) - g(x)} \right] = \frac{l + 2m}{2l - m} \)
- (c) \( \lim_{{x \to a}} \left[ 2f(x) + 3g(x) \right] = 2l + 3m \)
- (d) \( \lim_{{x \to a}} \left[ \frac{f(x) + g(x)}{(g(x))^2} \right] = \frac{l + m}{m^2} \)

Would you like a detailed explanation on any of these parts or have any other questions?

Here are some related questions:
1. What is the significance of the product and quotient rules in limit evaluation?
2. How does the limit change if \(l\) or \(m\) approaches zero?
3. Can you provide a real-world example where these types of limits are applied?
4. How does continuity of functions affect the evaluation of limits?
5. What happens if \(f(x)\) or \(g(x)\) is undefined at \(x = a\)?

**Tip:** When dealing with limits, it's important to consider the behavior of the function near the point of interest, not just at that specific point.

If lim (x→a) f(x) = l and lim (x→a) g(x) = m, evaluate the following limits: (a) lim (x→a) f(x)g(x), (b) lim (x→a) [ (f(x) + 2g(x)) / (2f(x) - g(x)) ], (c) lim (x→a) [ 2f(x) + 3g(x) ], (d) lim (x→a) [ (f(x) + g(x)) / (g(x))^2 ], when l, m > 0.

This problem involves evaluating limits of function combinations given the limits of individual functions as x approaches a. Concepts covered include the product rule, quotient rule, and sum rule for limits. Suitable for students in Grades 11-12.

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