given that\n\\( \\lim _{x \\rightarrow a} f(x)=0 \\)\n\\( \\lim _{x \\rightarrow a} g(x)=0 \\)\n\\( \\lim…

given that\n\\( \\lim _{x \\rightarrow a} f(x)=0 \\)\n\\( \\lim _{x \\rightarrow a} g(x)=0 \\)\n\\( \\lim _{x \\rightarrow a} h(x)=1 \\)\n\\( \\lim _{x \\rightarrow a} p(x)=\\infty \\)\n\\( \\lim _{x \\rightarrow a} q(x)=\\infty \\),\nevaluate if the following limits are not indeterminate forms. (if a limit is indeterminate, enter indeterminate.)\n(a) \\( \\lim _{x \\rightarrow a} \\frac{f(x)}{g(x)} \\)\n(b) \\( \\lim _{x \\rightarrow a} \\frac{f(x)}{p(x)} \\)\n(c) \\( \\lim _{x \\rightarrow a} \\frac{h(x)}{p(x)} \\)\n(d) \\( \\lim _{x \\rightarrow a} \\frac{p(x)}{q(x)} \\)

given that\n\\( \\lim _{x \\rightarrow a} f(x)=0 \\)\n\\( \\lim _{x \\rightarrow a} g(x)=0 \\)\n\\( \\lim _{x \\rightarrow a} h(x)=1 \\)\n\\( \\lim _{x \\rightarrow a} p(x)=\\infty \\)\n\\( \\lim _{x \\rightarrow a} q(x)=\\infty \\),\nevaluate if the following limits are not indeterminate forms. (if a limit is indeterminate, enter indeterminate.)\n(a) \\( \\lim _{x \\rightarrow a} \\frac{f(x)}{g(x)} \\)\n(b) \\( \\lim _{x \\rightarrow a} \\frac{f(x)}{p(x)} \\)\n(c) \\( \\lim _{x \\rightarrow a} \\frac{h(x)}{p(x)} \\)\n(d) \\( \\lim _{x \\rightarrow a} \\frac{p(x)}{q(x)} \\)

Answer

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\rightarrow a}f(x) = 0) and (\lim_{x\rightarrow a}g(x)=0). So (\lim_{x\rightarrow 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\rightarrow a}f(x) = 0) and (\lim_{x\rightarrow a}p(x)=\infty). Using the property (\lim_{x\rightarrow a}\frac{f(x)}{p(x)}=\frac{\lim_{x\rightarrow a}f(x)}{\lim_{x\rightarrow a}p(x)}) (since (\lim_{x\rightarrow a}p(x)\neq0)), we get (\frac{0}{\infty}=0).

Step4: Analyze part (c)

Given (\lim_{x\rightarrow a}h(x) = 1) and (\lim_{x\rightarrow a}p(x)=\infty). Using the property (\lim_{x\rightarrow a}\frac{h(x)}{p(x)}=\frac{\lim_{x\rightarrow a}h(x)}{\lim_{x\rightarrow a}p(x)}) (since (\lim_{x\rightarrow a}p(x)\neq0)), we have (\frac{1}{\infty}=0).

Step5: Analyze part (d)

Since (\lim_{x\rightarrow a}p(x)=\infty) and (\lim_{x\rightarrow a}q(x)=\infty), (\lim_{x\rightarrow 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