Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 163782 by bekzodjumayev last updated on 10/Jan/22

Commented by bekzodjumayev last updated on 10/Jan/22

Please help

$${Please}\:{help} \\ $$

Answered by Ar Brandon last updated on 10/Jan/22

L=lim_(x→−1) ((x^n −nx+(1−n))/((x+1)^2 ))      =lim_(t→0) (((t−1)^n −n(t−1)+(1−n))/t^2 ) , n odd⇒ (−1)^n =−1      =lim_(t→0) (((−1)^n (1−nt−((n(n−1))/2)t^2 )−nt+n+1−n)/t^2 )      =lim_(t→0) ((−1+nt+((n(n−1))/2)t^2 −nt+1)/t^2 )      =lim_(t→0) ((n(n−1)t^2 )/(2t^2 ))=((n(n−1))/2)

$$\mathcal{L}=\underset{{x}\rightarrow−\mathrm{1}} {\mathrm{lim}}\frac{{x}^{{n}} −{nx}+\left(\mathrm{1}−{n}\right)}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\:\:\:\:=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({t}−\mathrm{1}\right)^{{n}} −{n}\left({t}−\mathrm{1}\right)+\left(\mathrm{1}−{n}\right)}{{t}^{\mathrm{2}} }\:,\:{n}\:\mathrm{odd}\Rightarrow\:\left(−\mathrm{1}\right)^{{n}} =−\mathrm{1} \\ $$$$\:\:\:\:=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(−\mathrm{1}\right)^{{n}} \left(\mathrm{1}−{nt}−\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}}{t}^{\mathrm{2}} \right)−{nt}+{n}+\mathrm{1}−{n}}{{t}^{\mathrm{2}} } \\ $$$$\:\:\:\:=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\mathrm{1}+{nt}+\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}}{t}^{\mathrm{2}} −{nt}+\mathrm{1}}{{t}^{\mathrm{2}} } \\ $$$$\:\:\:\:=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{n}\left({n}−\mathrm{1}\right){t}^{\mathrm{2}} }{\mathrm{2}{t}^{\mathrm{2}} }=\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}} \\ $$

Commented by bekzodjumayev last updated on 11/Jan/22

Thank you

$${Thank}\:{you} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com