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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

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

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