https://file.aiquickdraw.com/mathbot/file/1729443762627zyxfg2px.jpg

Let's go through each limit provided in the image step by step, using the given information about the limits of functions \(f(x)\), \(g(x)\), \(h(x)\), \(p(x)\), and \(q(x)\):

Given:
- \(\lim\limits_{x \to a} f(x) = 0\),
- \(\lim\limits_{x \to a} g(x) = 0\),
- \(\lim\limits_{x \to a} h(x) = 1\),
- \(\lim\limits_{x \to a} p(x) = \infty\),
- \(\lim\limits_{x \to a} q(x) = \infty\).

### (a) \(\lim\limits_{x \to a} [f(x) - p(x)]\)

We know:
- \(\lim\limits_{x \to a} f(x) = 0\),
- \(\lim\limits_{x \to a} p(x) = \infty\).

Therefore:
\[
\lim\limits_{x \to a} [f(x) - p(x)] = 0 - \infty = -\infty.
\]

Hence, the result is \(\text{NINF}\) (negative infinity).

### (b) \(\lim\limits_{x \to a} [p(x) - q(x)]\)

We know:
- \(\lim\limits_{x \to a} p(x) = \infty\),
- \(\lim\limits_{x \to a} q(x) = \infty\).

Therefore:
\[
\lim\limits_{x \to a} [p(x) - q(x)] = \infty - \infty.
\]
This is an **indeterminate form**. Hence, the result is \(I\) (indeterminate).

### (c) \(\lim\limits_{x \to a} [p(x) + q(x)]\)

We know:
- \(\lim\limits_{x \to a} p(x) = \infty\),
- \(\lim\limits_{x \to a} q(x) = \infty\).

Therefore:
\[
\lim\limits_{x \to a} [p(x) + q(x)] = \infty + \infty = \infty.
\]

Hence, the result is \(\text{INF}\) (positive infinity).

### Final Answers:
- (a) NINF
- (b) \(I\) (indeterminate)
- (c) INF

Would you like further clarification or details on any step?

Here are some related questions for further exploration:
1. What are other common indeterminate forms in limits?
2. How can L'Hopital's Rule be applied to resolve \(\infty - \infty\) indeterminate forms?
3. What are the necessary conditions to apply L'Hopital's Rule?
4. How do limits behave when functions approach \(0/0\) or \(\infty/\infty\)?
5. Can the squeeze theorem help evaluate certain limits?

**Tip:** When encountering indeterminate forms like \( \infty - \infty \), applying algebraic manipulation or L'Hopital's Rule can often help resolve the limit.

Given the limits \( \lim_{x \to a} f(x) = 0 \), \( \lim_{x \to a} g(x) = 0 \), \( \lim_{x \to a} h(x) = 1 \), \( \lim_{x \to a} p(x) = \infty \), \( \lim_{x \to a} q(x) = \infty \), which of the following limits are determinate? (a) \( \lim_{x \to a} [f(x) - p(x)] \), (b) \( \lim_{x \to a} [p(x) - q(x)] \), (c) \( \lim_{x \to a} [p(x) + q(x)] \).

Explore how to evaluate limits involving indeterminate forms and infinity. This problem reviews common limit properties and indeterminate forms, focusing on expressions with \( \infty \) and 0. Ideal for advanced high school or undergraduate calculus students.

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