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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.

Compute the limits that exist, given that lim(x→0) f(x) = -1/2 and lim(x→0) g(x) = 1/2. Find the following limits:
a) lim(x→0) [f(x) + g(x)]
b) lim(x→0) [f(x) - 2g(x)]
c) lim(x→0) [f(x) * g(x)]
d) lim(x→0) [f(x) / g(x)]

Learn how to compute various limits using key calculus concepts. This problem includes the sum, product, and quotient of functions at a given limit, focusing on f(x) and g(x) with given values. Perfect for students learning limits in calculus.

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