question 16\n5 pts\nlet ( p(x) ) and ( q(x) ) be polynomials. find ( lim _{x \rightarrow infty}…

question 16\n5 pts\nlet ( p(x) ) and ( q(x) ) be polynomials. find ( lim _{x \rightarrow infty} \frac{p(x)}{q(x)} ) if the degree of ( p(x) ) is 5 and the degree of ( q(x) ) is 8.\n-8\n0\n5\nno correct answer choice is given.\n-5\n8

question 16\n5 pts\nlet ( p(x) ) and ( q(x) ) be polynomials. find ( lim _{x \rightarrow infty} \frac{p(x)}{q(x)} ) if the degree of ( p(x) ) is 5 and the degree of ( q(x) ) is 8.\n-8\n0\n5\nno correct answer choice is given.\n-5\n8

Answer

Explanation:

Step1: Recall the rule for limits of rational functions

For a rational function (\frac{P(x)}{Q(x)}) where (P(x)) and (Q(x)) are polynomials. If (\text{deg}(P(x))=m) and (\text{deg}(Q(x))=n)

  • If (m < n), then (\lim_{x\rightarrow\infty}\frac{P(x)}{Q(x)} = 0)
  • If (m=n), then (\lim_{x\rightarrow\infty}\frac{P(x)}{Q(x)}=\frac{a_m}{b_n}) (where (a_m) is the leading coefficient of (P(x)) and (b_n) is the leading coefficient of (Q(x)))
  • If (m>n), then (\lim_{x\rightarrow\infty}\frac{P(x)}{Q(x)}=\pm\infty)

Step2: Apply the rule to the given problem

Here, (\text{deg}(P(x)) = 5) and (\text{deg}(Q(x))=8). Since (5<8) (i.e., (m < n))

Answer:

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