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∵
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

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