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Question Number 69075 by Aditya789 last updated on 19/Sep/19

Commented by Prithwish sen last updated on 19/Sep/19

Divide both numerator and denominator  by x^3  and the limit will be 1

$$\mathrm{Divide}\:\mathrm{both}\:\mathrm{numerator}\:\mathrm{and}\:\mathrm{denominator} \\ $$$$\mathrm{by}\:\mathrm{x}^{\mathrm{3}} \:\mathrm{and}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{will}\:\mathrm{be}\:\mathrm{1} \\ $$

Commented by mathmax by abdo last updated on 19/Sep/19

(x−2)(x+3)(x+5) =x^3  +p(x) with deg(p)≤2  (x−1)(x−3)(x−7) =x^3  +q(x) with deg(a)≤2 ⇒  lim_(x→+^− ∞) (((x−2)(x+3)(x+5))/((x−1)(x−3)(x−7))) =lim_(x→+^− ∞)     ((x^3  +p(x))/(x^3  +q(x)))  =lim_(x→+^− ∞)  (x^3 /x^3 ) =1 .

$$\left({x}−\mathrm{2}\right)\left({x}+\mathrm{3}\right)\left({x}+\mathrm{5}\right)\:={x}^{\mathrm{3}} \:+{p}\left({x}\right)\:{with}\:{deg}\left({p}\right)\leqslant\mathrm{2} \\ $$$$\left({x}−\mathrm{1}\right)\left({x}−\mathrm{3}\right)\left({x}−\mathrm{7}\right)\:={x}^{\mathrm{3}} \:+{q}\left({x}\right)\:{with}\:{deg}\left({a}\right)\leqslant\mathrm{2}\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow\overset{−} {+}\infty} \frac{\left({x}−\mathrm{2}\right)\left({x}+\mathrm{3}\right)\left({x}+\mathrm{5}\right)}{\left({x}−\mathrm{1}\right)\left({x}−\mathrm{3}\right)\left({x}−\mathrm{7}\right)}\:={lim}_{{x}\rightarrow\overset{−} {+}\infty} \:\:\:\:\frac{{x}^{\mathrm{3}} \:+{p}\left({x}\right)}{{x}^{\mathrm{3}} \:+{q}\left({x}\right)} \\ $$$$={lim}_{{x}\rightarrow\overset{−} {+}\infty} \:\frac{{x}^{\mathrm{3}} }{{x}^{\mathrm{3}} }\:=\mathrm{1}\:. \\ $$

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