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lim-x-0-x-1-n-1-n-1-x-1-n-x-2-n-N-Calculate-Without-a-L-Hopital-s-rule-




Question Number 125292 by Kurbanklichevs last updated on 09/Dec/20
lim_(x→0)  (((x+1)^(n+1) −(n+1)(x+1)+n)/x^2 )     n∈N  Calculate. (Without a L′Hopital′s rule)
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left({x}+\mathrm{1}\right)^{{n}+\mathrm{1}} −\left({n}+\mathrm{1}\right)\left({x}+\mathrm{1}\right)+{n}}{{x}^{\mathrm{2}} }\:\:\:\:\:{n}\in\mathbb{N} \\ $$$${Calculate}.\:\left({Without}\:{a}\:{L}'{Hopital}'{s}\:{rule}\right) \\ $$
Answered by Dwaipayan Shikari last updated on 09/Dec/20
lim_(x→0) (((1+x)^(n+1) −(n+1)(x+1)+n)/x^2 )  lim_(x→0) ((1+(n+1)x+((n(n+1))/(2!))x^2 −(n+1)(x+1)+n)/x^2 )  lim_(x→0) ((1−n−1−n+((n(n+1))/(2!))x^2 )/x^2 )  =((n(n+1))/2)
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{1}+{x}\right)^{{n}+\mathrm{1}} −\left({n}+\mathrm{1}\right)\left({x}+\mathrm{1}\right)+{n}}{{x}^{\mathrm{2}} } \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}+\left({n}+\mathrm{1}\right){x}+\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}!}{x}^{\mathrm{2}} −\left({n}+\mathrm{1}\right)\left({x}+\mathrm{1}\right)+{n}}{{x}^{\mathrm{2}} } \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−{n}−\mathrm{1}−{n}+\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}!}{x}^{\mathrm{2}} }{{x}^{\mathrm{2}} } \\ $$$$=\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$
Commented by Kurbanklichevs last updated on 10/Dec/20
Tank you
$${Tank}\:{you} \\ $$

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