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Question Number 167434 by greogoury55 last updated on 16/Mar/22

$$\underset{x\to a}{\lim }\frac{x^n-a^n-{na}^{n-1}(x-a)}{{(x-a)}^2}=?$$

Answered by qaz last updated on 16/Mar/22

$$\underset{\mathrm{x}\to \mathrm{a}}{\lim }\frac{x^n-a^n-{na}^{n-1}(x-a)}{{(x-a)}^2}$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{{(\mathrm{x}+\mathrm{a})}^{\mathrm{n}}-{\mathrm{a}}^{\mathrm{n}}-{\mathrm{na}}^{\mathrm{n}-1}\mathrm{x}}{{\mathrm{x}}^2}$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{n}{(\mathrm{x}+\mathrm{a})}^{\mathrm{n}-1}-{\mathrm{na}}^{\mathrm{n}-1}}{2\mathrm{x}}$$$$=\frac{1}{2}\mathrm{n}(\mathrm{n}-1){\mathrm{a}}^{\mathrm{n}-2}$$

Answered by cortano1 last updated on 16/Mar/22

$$\underset{x\to a}{\lim }\frac{x^n-a^n-{na}^{n-1}(x-a)}{{(x-a)}^2}=$$$$\underset{x\to 0}{\lim }\frac{{(x+a)}^n-a^n-\frac{{na}^n}{a}x}{x^2}=$$$$a^n\times \underset{x\to 0}{\lim }\frac{{(1+\frac{x}{a})}^n-1-\frac{nx}{a}}{x^2}=$$$$a^n\times \underset{x\to 0}{\lim }\frac{1+\frac{nx}{a}+\frac{n(n-1)}{2}{\left(\frac{x}{a}\right)}^2-1-\frac{nx}{a}}{x^2}=$$$$a^n\times \underset{x\to 0}{\lim }\frac{\frac{n(n-1)}{2}.\frac{x^2}{a^2}}{x^2}=\frac{n(n-1)a^n}{2a^2}$$$$=\frac{1}{2}n(n-1)a^{n-2}$$

Commented by greogoury55 last updated on 18/Mar/22

$$nice$$

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