Let's analyze the problem and evaluate the limits provided:

Given limits:

• $\lim_{{x \to a}} f(x) = 0$
• $\lim_{{x \to a}} g(x) = 0$
• $\lim_{{x \to a}} h(x) = 1$
• $\lim_{{x \to a}} p(x) = \infty$
• $\lim_{{x \to a}} q(x) = \infty$

We need to classify the following limits:

(a) $\lim_{{x \to a}} [f(x) - p(x)]$

• As $f(x) \to 0$ and $p(x) \to \infty$, this results in the form $0 - \infty$.
• A limit in the form $0 - \infty$ is not indeterminate because subtracting any finite number (0) from $\infty$ results in $-\infty$.
• Thus, this limit evaluates to $-\infty$.

(b) $\lim_{{x \to a}} [p(x) - q(x)]$

• Both $p(x)$ and $q(x)$ tend to $\infty$, so this limit is in the form $\infty - \infty$.
• The form $\infty - \infty$ is indeterminate, as the actual behavior depends on how $p(x)$ and $q(x)$ approach infinity relative to each other.
• Thus, this limit is indeterminate ($I$).

(c) $\lim_{{x \to a}} [p(x) + q(x)]$

• Both $p(x)$ and $q(x)$ tend to $\infty$, so this limit is in the form $\infty + \infty$.
• A limit in the form $\infty + \infty$ is not indeterminate and will result in $\infty$.
• Thus, this limit evaluates to $\infty$ ($INF$).

Final Answers:

• (a) $-\infty$
• (b) $I$ (Indeterminate)
• (c) $INF$ ($\infty$)

Would you like more details or explanations on any part of this? Here are a few related questions to deepen your understanding:

1. What other types of indeterminate forms exist beyond $\infty - \infty$?
2. How do you apply L'Hôpital's rule to evaluate indeterminate limits like $\infty - \infty$?
3. Why is the form $0 \cdot \infty$ considered indeterminate?
4. How does the behavior of $p(x)$ and $q(x)$ affect the outcome of the limit in (b)?
5. Can indeterminate forms always be resolved using algebraic manipulation?

Tip: When evaluating limits, always check if the form is indeterminate before applying advanced techniques like L'Hôpital's rule.