Question

$$\lim_{x \to a} f(x)=0$$ $$\lim_{x \to a} g(x)=0$$ $$\lim_{x \to a} h(x)=1$$ $$\lim_{x \to a} p(x)=\infty$$ $$\lim_{x \to a} q(x)=\infty,$$ evaluate if the following limits not are indeterminate forms. (if a limit is indeterminate, enter indeterminate.)
(a) $$\lim_{x \to a}f(x)^{g(x)}$$
(b) $$\lim_{x \to a}f(x)^{p(x)}$$
(c) $$\lim_{x \to a}h(x)^{p(x)}$$
(d) $$\lim_{x \to a}p(x)^{f(x)}$$
(e) $$\lim_{x \to a}p(x)^{q(x)}$$
(f) $$\lim_{x \to a} \sqrtq(x){p(x)}$$

Explanation:

Step1: Recall indeterminate forms

Indeterminate forms include \(0^{0}\), \(\infty^{0}\), \(1^{\infty}\), \(0\times\infty\), \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(\infty-\infty\)

Step2: Evaluate part (a)

We have \(\lim_{x
ightarrow a}f(x) = 0\) and \(\lim_{x
ightarrow a}g(x)=0\). The form \(\lim_{x
ightarrow a}[f(x)]^{g(x)}\) is of the \(0^{0}\) indeterminate form.

Step3: Evaluate part (b)

Since \(\lim_{x
ightarrow a}f(x) = 0\) and \(\lim_{x
ightarrow a}p(x)=\infty\), the form \(\lim_{x
ightarrow a}[f(x)]^{p(x)}\) is of the \(0^{\infty}\) form. \(0^{\infty}=0\) (not indeterminate)

Step4: Evaluate part (c)

Given \(\lim_{x
ightarrow a}h(x) = 1\) and \(\lim_{x
ightarrow a}p(x)=\infty\), the form \(\lim_{x
ightarrow a}[h(x)]^{p(x)}\) is of the \(1^{\infty}\) indeterminate form.

Step5: Evaluate part (d)

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

Step6: Evaluate part (e)

Given \(\lim_{x
ightarrow a}p(x)=\infty\) and \(\lim_{x
ightarrow a}q(x)=\infty\), the form \(\lim_{x
ightarrow a}[p(x)]^{q(x)}\) is of the \(\infty^{\infty}\) form. \(\infty^{\infty}=\infty\) (not indeterminate)

Step7: Evaluate part (f)

We can rewrite \(\lim_{x
ightarrow a}\sqrt[q(x)]{p(x)}=\lim_{x
ightarrow a}[p(x)]^{\frac{1}{q(x)}}\). Since \(\lim_{x
ightarrow a}p(x)=\infty\) and \(\lim_{x
ightarrow a}q(x)=\infty\), then \(\lim_{x
ightarrow a}\frac{1}{q(x)} = 0\). The form \(\lim_{x
ightarrow a}[p(x)]^{\frac{1}{q(x)}}\) is of the \(\infty^{0}\) indeterminate form.

Answer:

(a) INDETERMINATE
(b) \(0\)
(c) INDETERMINATE
(d) INDETERMINATE
(e) \(\infty\)
(f) INDETERMINATE