找出P和Q的值并使得x→∞时f(x)=(px^2-2)/(x^2+1)+3qx+5有无穷小

Question

找出P和Q的值并使得x→∞ 时f(x)=(px^2-2)/(x^2+1)+3qx+5有无穷小

Answer 1

∵
lim(x→∞) (px^2-2)/(x^2+1)+3qx+5 = 0
∴①
0 = lim(x→∞) { (px^2-2)/(x^2+1)+3qx+5 }/x
= lim(x→∞) (px-2/x)/(x^2+1)+3q +5/x
= lim(x→∞) (p/x-2/x^3)/(1+1/x^2) + 3q + 5/x
= 0 + 3q + 0
=3q
∴ q = 0
②
∵
0 = lim(x→∞) (px^2-2)/(x^2+1) + 5
= lim(x→∞) (p-2/x^2)/(1+1/x^2) + 5
= p + 5
∴ p = -5