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}f(x) p(x) \\)\n(b) \\( \\lim _{x \\rightarrow a}h(x) p(x) \\)\n(c) \\( \\lim _{x \\rightarrow a}p(x) q(x) \\)
Answer
Explanation:
(a) $\lim_{x\rightarrow a}[f(x)p(x)]$
- Step1: Recall indeterminate forms The form (0\times\infty) is an indeterminate form. Since (\lim_{x\rightarrow a}f(x) = 0) and (\lim_{x\rightarrow a}p(x)=\infty), the limit (\lim_{x\rightarrow a}[f(x)p(x)]) is of the form (0\times\infty).
(b) (\lim_{x\rightarrow a}[h(x)p(x)])
- Step1: Use limit - product rule (in a non - indeterminate sense) We know that if (\lim_{x\rightarrow a}h(x)=L) (where (L\neq0)) and (\lim_{x\rightarrow a}p(x)=\infty), then (\lim_{x\rightarrow a}[h(x)p(x)]=\infty) when (L > 0) and (-\infty) when (L<0). Here, (\lim_{x\rightarrow a}h(x) = 1>0) and (\lim_{x\rightarrow a}p(x)=\infty). By the properties of limits, if (\lim_{x\rightarrow a}h(x)=1) and (\lim_{x\rightarrow a}p(x)=\infty), then (\lim_{x\rightarrow a}[h(x)p(x)]=\infty) (not an indeterminate form).
(c) (\lim_{x\rightarrow a}[p(x)q(x)])
- Step1: Use limit - product rule (in a non - indeterminate sense) If (\lim_{x\rightarrow a}p(x)=\infty) and (\lim_{x\rightarrow 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\rightarrow a}[p(x)q(x)]=\infty) (if we consider the case where (p(x)) and (q(x)) approach (-\infty), (\lim_{x\rightarrow a}[p(x)q(x)]=\infty) as ((-\infty)\times(-\infty)=\infty)). This is not an indeterminate form.
Answer:
(a) INDETERMINATE (b) (\infty) (c) (\infty)