Answer
(a): The limit is not indeterminate. Its value is $-\infty$.
(b): The limit is indeterminate, in the form of $\infty-\infty$.
(c): The limit is not indeterminate. Its value is $\infty$.
Work Step by Step
(a): We are asked to determine if the following is in an indeterminate form:
$$\lim\limits_{x \to a} [f(x) - p(x)]$$
Given that $\lim\limits_{x \to a} f(x) = 0$ and $\lim\limits_{x \to a} p(x)=\infty$, we can say that the limit is not indeterminate. $0-\infty$ is equivalent to $-\infty$, therefore the limit is $-\infty$
(b) $$\lim\limits_{x \to a} [p(x) - q(x)]$$
Given that $\lim\limits_{x \to a} p(x)=\infty$ and $\lim\limits_{x \to a} q(x) = \infty$, we can say that this is an indeterminate form. We cannot evaluate this limit without more information because the limit as it stands now is $\infty - \infty$.
(c) $$\lim\limits_{x \to a} [p(x) + q(x)]$$
Given that $\lim\limits_{x \to a} p(x)=\infty$ and $\lim\limits_{x \to a} q(x) = \infty$, we can say that this is not an indeterminate form. $\infty+\infty$ is simply $\infty$.
