Question

given that
\\( \lim _{x \
ightarrow a} f(x)=0 \\)
\\( \lim _{x \
ightarrow a} g(x)=0 \\)
\\( \lim _{x \
ightarrow a} h(x)=1 \\)
\\( \lim _{x \
ightarrow a} p(x)=\infty \\)
\\( \lim _{x \
ightarrow a} q(x)=\infty \\),
evaluate if the following limits are not indeterminate forms. (if a limit is indeterminate, enter indeterminate.)
(a) \\( \lim _{x \
ightarrow a}f(x)-p(x) \\)
(b) \\( \lim _{x \
ightarrow a}p(x)-q(x) \\)
(c) \\( \lim _{x \
ightarrow a}p(x)+q(x) \\)

Explanation:

Step1: Analyze \(\lim_{x

ightarrow a}[f(x)-p(x)]\)
We know that \(\lim_{x
ightarrow a}f(x) = 0\) and \(\lim_{x
ightarrow a}p(x)=\infty\).
Using the property of limits \(\lim_{x
ightarrow a}[f(x)-p(x)]=\lim_{x
ightarrow a}f(x)-\lim_{x
ightarrow a}p(x)\)
Substitute the values: \(0-\infty=-\infty\)

Step2: Analyze \(\lim_{x

ightarrow a}[p(x)-q(x)]\)
Since \(\lim_{x
ightarrow a}p(x)=\infty\) and \(\lim_{x
ightarrow a}q(x)=\infty\).
The form \(\infty-\infty\) is an indeterminate form.

Step3: Analyze \(\lim_{x

ightarrow a}[p(x)+q(x)]\)
Given \(\lim_{x
ightarrow a}p(x)=\infty\) and \(\lim_{x
ightarrow a}q(x)=\infty\)
Using the property of limits \(\lim_{x
ightarrow a}[p(x)+q(x)]=\lim_{x
ightarrow a}p(x)+\lim_{x
ightarrow a}q(x)\)
Substitute the values: \(\infty+\infty=\infty\)

Answer:

(a) \(-\infty\)
(b) INDETERMINATE
(c) \(\infty\)