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

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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Geraldine Flores

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 $$\displaystyle-\infty$$