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} \frac{f(x)}{g(x)} \\)
(b) \\( \lim _{x \
ightarrow a} \frac{f(x)}{p(x)} \\)
(c) \\( \lim _{x \
ightarrow a} \frac{h(x)}{p(x)} \\)
(d) \\( \lim _{x \
ightarrow a} \frac{p(x)}{q(x)} \\)

Explanation:

Step1: Recall indeterminate forms

The indeterminate forms are \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0\times\infty\), \(\infty-\infty\), \(0^{0}\), \(\infty^{0}\), \(1^{\infty}\).

Step2: Analyze part (a)

We have \(\lim_{x
ightarrow a}f(x) = 0\) and \(\lim_{x
ightarrow a}g(x)=0\). So \(\lim_{x
ightarrow a}\frac{f(x)}{g(x)}\) is of the form \(\frac{0}{0}\), which is an indeterminate form.

Step3: Analyze part (b)

We know that \(\lim_{x
ightarrow a}f(x) = 0\) and \(\lim_{x
ightarrow a}p(x)=\infty\). Using the property \(\lim_{x
ightarrow a}\frac{f(x)}{p(x)}=\frac{\lim_{x
ightarrow a}f(x)}{\lim_{x
ightarrow a}p(x)}\) (since \(\lim_{x
ightarrow a}p(x)
eq0\)), we get \(\frac{0}{\infty}=0\).

Step4: Analyze part (c)

Given \(\lim_{x
ightarrow a}h(x) = 1\) and \(\lim_{x
ightarrow a}p(x)=\infty\). Using the property \(\lim_{x
ightarrow a}\frac{h(x)}{p(x)}=\frac{\lim_{x
ightarrow a}h(x)}{\lim_{x
ightarrow a}p(x)}\) (since \(\lim_{x
ightarrow a}p(x)
eq0\)), we have \(\frac{1}{\infty}=0\).

Step5: Analyze part (d)

Since \(\lim_{x
ightarrow a}p(x)=\infty\) and \(\lim_{x
ightarrow a}q(x)=\infty\), \(\lim_{x
ightarrow a}\frac{p(x)}{q(x)}\) is of the form \(\frac{\infty}{\infty}\), which is an indeterminate form.

Answer:

(a) INDETERMINATE
(b) \(0\)
(c) \(0\)
(d) INDETERMINATE