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.