Let's compute the given limits step by step. The problem provides us with the following information:

$\lim_{x \to 0} f(x) = -\frac{1}{2} \quad \text{and} \quad \lim_{x \to 0} g(x) = \frac{1}{2}$

We will apply limit properties to solve each part of the question.

a) $\lim_{x \to 0} \left[f(x) + g(x)\right]$

Using the sum rule for limits: $\lim_{x \to 0} \left[f(x) + g(x)\right] = \lim_{x \to 0} f(x) + \lim_{x \to 0} g(x)$

Substitute the given limits: $\lim_{x \to 0} \left[f(x) + g(x)\right] = -\frac{1}{2} + \frac{1}{2} = 0$

b) $\lim_{x \to 0} \left[f(x) - 2g(x)\right]$

Using the difference and scalar multiplication properties of limits: $\lim_{x \to 0} \left[f(x) - 2g(x)\right] = \lim_{x \to 0} f(x) - 2\lim_{x \to 0} g(x)$

Substitute the given limits: $\lim_{x \to 0} \left[f(x) - 2g(x)\right] = -\frac{1}{2} - 2\left(\frac{1}{2}\right) = -\frac{1}{2} - 1 = -\frac{3}{2}$

c) $\lim_{x \to 0} \left[f(x) \times g(x)\right]$

Using the product rule for limits: $\lim_{x \to 0} \left[f(x) \times g(x)\right] = \lim_{x \to 0} f(x) \times \lim_{x \to 0} g(x)$

Substitute the given limits: $\lim_{x \to 0} \left[f(x) \times g(x)\right] = -\frac{1}{2} \times \frac{1}{2} = -\frac{1}{4}$

d) $\lim_{x \to 0} \frac{f(x)}{g(x)}$

Using the quotient rule for limits, and noting that $\lim_{x \to 0} g(x) \neq 0$: $\lim_{x \to 0} \frac{f(x)}{g(x)} = \frac{\lim_{x \to 0} f(x)}{\lim_{x \to 0} g(x)}$

Substitute the given limits: $\lim_{x \to 0} \frac{f(x)}{g(x)} = \frac{-\frac{1}{2}}{\frac{1}{2}} = -1$

Summary of the limits:

• a) $\lim_{x \to 0} \left[f(x) + g(x)\right] = 0$
• b) $\lim_{x \to 0} \left[f(x) - 2g(x)\right] = -\frac{3}{2}$
• c) $\lim_{x \to 0} \left[f(x) \times g(x)\right] = -\frac{1}{4}$
• d) $\lim_{x \to 0} \frac{f(x)}{g(x)} = -1$

Would you like any further clarification or more details on any of these steps?

Here are 5 related questions for practice:

1. What is the limit of $f(x) + 3g(x)$ as $x \to 0$?
2. Compute $\lim_{x \to 0} \left[2f(x) - 3g(x)\right]$ using the same given limits.
3. What is the result of $\lim_{x \to 0} \left[\frac{f(x) \times g(x)}{g(x)}\right]$?
4. How would you find $\lim_{x \to 0} \left[f(x)^2 - g(x)^2\right]$?
5. Compute $\lim_{x \to 0} \left[3f(x) \cdot g(x)\right]$ using the limits provided.

Tip: Remember to use limit laws like the sum, difference, product, and quotient rules for simplifying complex expressions.