Let P and Q be polynomials. Find limx→∞P(x)Q(x) if the degree of P is greater than the degree of Q.

Let P(X)=anxn+an−1xn−1+…+a1x+a0 and Q(x)=bmxm+bm−1xm−1+…+b1x+b0 where an≠0,bm≠0 and 0≤m&lt;n limx→∞P(x)Q(x)=limx→∞anxn+an−1xn−1+…+a1x+a0bmxm+bm−1xm−1+…+b1x+b0 =limx→∞anxnbmxm (when x→∞, only the leading term of the polynomial matters) =limx→∞anxn−mbm =limx→∞anbmxn−m =limx→∞anbm⋅∞ =∞ or −∞ (as which it should be, it all has to do with the signs of an and bm Result: ∞ or −∞

Solved: "Let P and Q be polynomials. Find limx→∞P(x)Q(x)..."

klytamnestra9a

Answered question

2021-11-12

Let P and Q be polynomials. Find

lim_{x \to \infty} \frac{P (x)}{Q (x)}

if the degree of P is greater than the degree of Q.

Answer & Explanation

Geraldine Flores

Beginner2021-11-13Added 21 answers

Let $P (X) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$ and $Q (x) = b_{m} x^{m} + b_{m - 1} x^{m - 1} + \dots + b_{1} x + b_{0}$ where $a_{n} \neq 0, b_{m} \neq 0$ and $0 \leq m < n$
$lim_{x \to \infty} \frac{P (x)}{Q (x)} = lim_{x \to \infty} \frac{a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}}{b_{m} x^{m} + b_{m - 1} x^{m - 1} + \dots + b_{1} x + b_{0}}$
$= lim_{x \to \infty} \frac{a_{n} x^{n}}{b_{m} x^{m}}$
(when $x \to \infty$ , only the leading term of the polynomial matters)
$= lim_{x \to \infty} \frac{a_{n} x^{n - m}}{b_{m}}$
$= lim_{x \to \infty} \frac{a_{n}}{b_{m}} x^{n - m}$
$= lim_{x \to \infty} \frac{a_{n}}{b_{m}} \cdot \infty$
$= \infty$ or $- \infty$
(as which it should be, it all has to do with the signs of $a_{n}$ and $b_{m}$
Result: $\infty$ or $- \infty$

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