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

     lim_(x→a)  ((x^n −a^n −na^(n−1) (x−a))/((x−a)^2 ))=?

$$\:\:\:\:\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\frac{{x}^{{n}} −{a}^{{n}} −{na}^{{n}−\mathrm{1}} \left({x}−{a}\right)}{\left({x}−{a}\right)^{\mathrm{2}} }=? \\ $$

Answered by qaz last updated on 16/Mar/22

lim_(x→a)  ((x^n −a^n −na^(n−1) (x−a))/((x−a)^2 ))  =lim_(x→0) (((x+a)^n −a^n −na^(n−1) x)/x^2 )  =lim_(x→0) ((n(x+a)^(n−1) −na^(n−1) )/(2x))  =(1/2)n(n−1)a^(n−2)

$$\underset{\mathrm{x}\rightarrow\mathrm{a}} {\mathrm{lim}}\:\frac{{x}^{{n}} −{a}^{{n}} −{na}^{{n}−\mathrm{1}} \left({x}−{a}\right)}{\left({x}−{a}\right)^{\mathrm{2}} } \\ $$$$=\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{x}+\mathrm{a}\right)^{\mathrm{n}} −\mathrm{a}^{\mathrm{n}} −\mathrm{na}^{\mathrm{n}−\mathrm{1}} \mathrm{x}}{\mathrm{x}^{\mathrm{2}} } \\ $$$$=\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{n}\left(\mathrm{x}+\mathrm{a}\right)^{\mathrm{n}−\mathrm{1}} −\mathrm{na}^{\mathrm{n}−\mathrm{1}} }{\mathrm{2x}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{n}\left(\mathrm{n}−\mathrm{1}\right)\mathrm{a}^{\mathrm{n}−\mathrm{2}} \\ $$

Answered by cortano1 last updated on 16/Mar/22

  lim_(x→a)  ((x^n −a^n −na^(n−1) (x−a))/((x−a)^2 )) =   lim_(x→0) (((x+a)^n −a^n −((na^n )/a)x)/x^2 ) =   a^n ×lim_(x→0)  (((1+(x/a))^n −1−((nx)/a))/x^2 ) =   a^n ×lim_(x→0)  ((1+((nx)/a)+((n(n−1))/2)((x/a))^2 −1−((nx)/a))/x^2 ) =   a^n ×lim_(x→0)  ((((n(n−1))/2).(x^2 /a^2 ))/x^2 ) = ((n(n−1)a^n )/(2a^2 ))   = (1/2)n(n−1)a^(n−2)

$$\:\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\frac{{x}^{{n}} −{a}^{{n}} −{na}^{{n}−\mathrm{1}} \left({x}−{a}\right)}{\left({x}−{a}\right)^{\mathrm{2}} }\:= \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({x}+{a}\right)^{{n}} −{a}^{{n}} −\frac{{na}^{{n}} }{{a}}{x}}{{x}^{\mathrm{2}} }\:= \\ $$$$\:{a}^{{n}} ×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}+\frac{{x}}{{a}}\right)^{{n}} −\mathrm{1}−\frac{{nx}}{{a}}}{{x}^{\mathrm{2}} }\:= \\ $$$$\:{a}^{{n}} ×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}+\frac{{nx}}{{a}}+\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}}\left(\frac{{x}}{{a}}\right)^{\mathrm{2}} −\mathrm{1}−\frac{{nx}}{{a}}}{{x}^{\mathrm{2}} }\:= \\ $$$$\:{a}^{{n}} ×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}}.\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }}{{x}^{\mathrm{2}} }\:=\:\frac{{n}\left({n}−\mathrm{1}\right){a}^{{n}} }{\mathrm{2}{a}^{\mathrm{2}} } \\ $$$$\:=\:\frac{\mathrm{1}}{\mathrm{2}}{n}\left({n}−\mathrm{1}\right){a}^{{n}−\mathrm{2}} \: \\ $$

Commented by greogoury55 last updated on 18/Mar/22

nice

$${nice} \\ $$

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