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Question Number 140384 by Willson last updated on 07/May/21

Answered by benjo_mathlover last updated on 07/May/21

(2) lim_(x→0) (((x^p +1)^n (x^q +1)^m −1)/x) = lim_(x→0) (((1+nx^p )(1+mx^q )−1)/x) = lim_(x→0) ((1+mx^q +nx^p +mnx^(p+q) −1)/x) = lim_(x→0) ((mx^q +nx^p +mnx^(p+q) )/x) = lim_(x→0) mx^(q−1) +nx^(p−1) +mnx^(p+q−1) = 0

$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{x}^{\mathrm{p}} +\mathrm{1}\right)^{\mathrm{n}} \left(\mathrm{x}^{\mathrm{q}} +\mathrm{1}\right)^{\mathrm{m}} −\mathrm{1}}{\mathrm{x}}\:= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}+\mathrm{nx}^{\mathrm{p}} \right)\left(\mathrm{1}+\mathrm{mx}^{\mathrm{q}} \right)−\mathrm{1}}{\mathrm{x}}\:= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}+\mathrm{mx}^{\mathrm{q}} +\mathrm{nx}^{\mathrm{p}} +\mathrm{mnx}^{\mathrm{p}+\mathrm{q}} −\mathrm{1}}{\mathrm{x}}\:= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{mx}^{\mathrm{q}} +\mathrm{nx}^{\mathrm{p}} +\mathrm{mnx}^{\mathrm{p}+\mathrm{q}} }{\mathrm{x}}\:= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{mx}^{\mathrm{q}−\mathrm{1}} +\mathrm{nx}^{\mathrm{p}−\mathrm{1}} +\mathrm{mnx}^{\mathrm{p}+\mathrm{q}−\mathrm{1}} \:=\:\mathrm{0} \\ $$

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