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The problem you uploaded provides two limit conditions:

1. \(\lim_{x \to 0} \left( f(x) + \frac{1}{g(x)} \right) = 4\)
2. \(\lim_{x \to 0} \left( f(x) - \frac{1}{g(x)} \right) = -3\)

You are asked to find:
\[
\lim_{x \to 0} \left( (f(x))^2 + \left( \frac{1}{g(x)} \right)^2 \right)
\]

### Step-by-step Solution:

Let's define:
\[
L_1 = \lim_{x \to 0} \left( f(x) + \frac{1}{g(x)} \right) = 4
\]
\[
L_2 = \lim_{x \to 0} \left( f(x) - \frac{1}{g(x)} \right) = -3
\]

Now, to find \( f(x) \) and \( \frac{1}{g(x)} \), we add and subtract these two equations:

1. **Add the equations:**
\[
L_1 + L_2 = \left( f(x) + \frac{1}{g(x)} \right) + \left( f(x) - \frac{1}{g(x)} \right)
\]
\[
4 + (-3) = 2f(x)
\]
\[
f(x) = \frac{1}{2}
\]

2. **Subtract the equations:**
\[
L_1 - L_2 = \left( f(x) + \frac{1}{g(x)} \right) - \left( f(x) - \frac{1}{g(x)} \right)
\]
\[
4 - (-3) = 2 \cdot \frac{1}{g(x)}
\]
\[
\frac{1}{g(x)} = \frac{7}{2}
\]

Now, substitute these values into the expression you are asked to find:

\[
\lim_{x \to 0} \left( (f(x))^2 + \left( \frac{1}{g(x)} \right)^2 \right)
\]

Using \( f(x) = \frac{1}{2} \) and \( \frac{1}{g(x)} = \frac{7}{2} \):

\[
(f(x))^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4}
\]
\[
\left( \frac{1}{g(x)} \right)^2 = \left( \frac{7}{2} \right)^2 = \frac{49}{4}
\]

Thus, the result is:
\[
\frac{1}{4} + \frac{49}{4} = \frac{50}{4} = 12.5
\]

### Final Answer:
\[
\lim_{x \to 0} \left( (f(x))^2 + \left( \frac{1}{g(x)} \right)^2 \right) = 12.5
\]

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Do you need further details or explanations?

Here are some related questions to explore:
1. What happens to the limit if one of the functions approaches infinity?
2. Can this method be applied if the limits involve indeterminate forms?
3. How would the result change if different limits were given?
4. What are the conditions for limits to exist in general?
5. How do we apply L'Hopital's rule in such limit problems?

**Tip:** When working with limits, always check for indeterminate forms like \( \frac{0}{0} \) or \( \infty - \infty \), as these may require special techniques like factoring or using L'Hopital's Rule.

Jika \( \lim_{x \to 0} \left( f(x) + \frac{1}{g(x)} \right) = 4 \) dan \( \lim_{x \to 0} \left( f(x) - \frac{1}{g(x)} \right) = -3 \), maka \( \lim_{x \to 0} \left( (f(x))^2 + \left( \frac{1}{g(x)} \right)^2 \right) \) adalah?

Explore how to solve the limit problem involving functions f(x) and g(x) where \( \lim_{x \to 0} \left( f(x) + \frac{1}{g(x)} \right) = 4 \) and \( \lim_{x \to 0} \left( f(x) - \frac{1}{g(x)} \right) = -3 \). Learn how to find the limit of the square of these terms.

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