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}h(x) p(x) \\)
(c) \\( \lim _{x \
ightarrow a}p(x) q(x) \\)

Explanation:

(a) $\lim_{x

ightarrow a}[f(x)p(x)]$

• Step1: Recall indeterminate forms

The form \(0\times\infty\) is an indeterminate form. Since \(\lim_{x
ightarrow a}f(x) = 0\) and \(\lim_{x
ightarrow a}p(x)=\infty\), the limit \(\lim_{x
ightarrow a}[f(x)p(x)]\) is of the form \(0\times\infty\).

(b) \(\lim_{x

ightarrow a}[h(x)p(x)]\)

• Step1: Use limit - product rule (in a non - indeterminate sense)

We know that if \(\lim_{x
ightarrow a}h(x)=L\) (where \(L
eq0\)) and \(\lim_{x
ightarrow a}p(x)=\infty\), then \(\lim_{x
ightarrow a}[h(x)p(x)]=\infty\) when \(L > 0\) and \(-\infty\) when \(L<0\). Here, \(\lim_{x
ightarrow a}h(x) = 1>0\) and \(\lim_{x
ightarrow a}p(x)=\infty\).
By the properties of limits, if \(\lim_{x
ightarrow a}h(x)=1\) and \(\lim_{x
ightarrow a}p(x)=\infty\), then \(\lim_{x
ightarrow a}[h(x)p(x)]=\infty\) (not an indeterminate form).

(c) \(\lim_{x

ightarrow a}[p(x)q(x)]\)

• Step1: Use limit - product rule (in a non - indeterminate sense)

If \(\lim_{x
ightarrow a}p(x)=\infty\) and \(\lim_{x
ightarrow a}q(x)=\infty\), then by the properties of limits for positive - infinity (if we consider the case where \(p(x)\) and \(q(x)\) approach \(+\infty\)) \(\lim_{x
ightarrow a}[p(x)q(x)]=\infty\) (if we consider the case where \(p(x)\) and \(q(x)\) approach \(-\infty\), \(\lim_{x
ightarrow a}[p(x)q(x)]=\infty\) as \((-\infty)\times(-\infty)=\infty\)). This is not an indeterminate form.

Answer:

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