Let P and Q be polynomials. Find \lim_{x\to\infty}\frac{P(x)}{Q(x)} if the

klytamnestra9a 2021-11-12 Answered
Let P and Q be polynomials. Find \(\displaystyle\lim_{{{x}\to\infty}}{\frac{{{P}{\left({x}\right)}}}{{{Q}{\left({x}\right)}}}}\) if the degree of P is greater than the degree of Q.

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Expert Answer

Geraldine Flores
Answered 2021-11-13 Author has 9487 answers

Let \(\displaystyle{P}{\left({X}\right)}={a}_{{n}}{x}^{{n}}+{a}_{{{n}-{1}}}{x}^{{{n}-{1}}}+\ldots+{a}_{{1}}{x}+{a}_{{0}}\) and \(\displaystyle{Q}{\left({x}\right)}={b}_{{m}}{x}^{{m}}+{b}_{{{m}-{1}}}{x}^{{{m}-{1}}}+\ldots+{b}_{{1}}{x}+{b}_{{0}}\) where \(\displaystyle{a}_{{n}}\ne{0},{b}_{{m}}\ne{0}\) and \(\displaystyle{0}\leq{m}{<}{n}\)
\(\displaystyle\lim_{{{x}\to\infty}}{\frac{{{P}{\left({x}\right)}}}{{{Q}{\left({x}\right)}}}}=\lim_{{{x}\to\infty}}{\frac{{{a}_{{n}}{x}^{{n}}+{a}_{{{n}-{1}}}{x}^{{{n}-{1}}}+\ldots+{a}_{{1}}{x}+{a}_{{0}}}}{{{b}_{{m}}{x}^{{m}}+{b}_{{{m}-{1}}}{x}^{{{m}-{1}}}+\ldots+{b}_{{1}}{x}+{b}_{{0}}}}}\)
\(\displaystyle=\lim_{{{x}\to\infty}}{\frac{{{a}_{{n}}{x}^{{n}}}}{{{b}_{{m}}{x}^{{m}}}}}\)
(when \(\displaystyle{x}\to\infty\), only the leading term of the polynomial matters)
\(\displaystyle=\lim_{{{x}\to\infty}}{\frac{{{a}_{{n}}{x}^{{{n}-{m}}}}}{{{b}_{{m}}}}}\)
\(\displaystyle=\lim_{{{x}\to\infty}}{\frac{{{a}_{{n}}}}{{{b}_{{m}}}}}{x}^{{{n}-{m}}}\)
\(\displaystyle=\lim_{{{x}\to\infty}}{\frac{{{a}_{{n}}}}{{{b}_{{m}}}}}\cdot\infty\)
\(\displaystyle=\infty\) or \(\displaystyle-\infty\)
(as which it should be, it all has to do with the signs of \(\displaystyle{a}_{{n}}\) and \(\displaystyle{b}_{{m}}\)
Result: \(\displaystyle\infty\) or \(\displaystyle-\infty\)

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