InterviewSolution
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InterviewSolution
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`lim_(xtoa) (x^(m)-a^(m))/(x-a)` का मान ज्ञात कीजिये। |
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Answer» `underset(xtoa)lim(x^(m)-a^(m))/(x-a)` मान `x=a+h` यदि `x=a,` तब `h=0" "therefore x to a implieshto 0` `therefore underset(xtoa)lim (x^(m)-a^(m))/(x-a)=underset(hto0)lim((a+h)^(m)-a^(m))/(a+h-a)` `=underset(hto0)lim(a^(m)(1+(h)/(a))^(m)-a^(m))/(h)` `=underset(hto0)lim(a^(m))/(h)[(1+(h)/(a))^(m)-1]` `=underset(hto0)lim (a^(m))/(h)[1+m((h)/(a))+(m(m-1))/(2!)((h)/(a))+...-1]` (द्विपद प्रमेय से) `=underset(hto0)lim (a^(m))/(h)[m((h)/(a))+(m(m-1))/(2!)((h)/(a))+...]` `=underset(xto0)lim(a^(m))/(h)m((h)/(a))[1+((m-1))/(2!)((h)/(a))+...]` `=underset(hto0)lim m a^(m-1)[1+((m-1))/(2!)((h)/(a))+...]` `=m*a^(m-1)[1+0+0+...]` (सीमा लेने पर) `=ma^(m-1)` |
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