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.