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Question Number 141699 by cesarL last updated on 22/May/21

If lim_(x→1) =(((x)^(1/k) −1)/(x−1))=L≠0 Find lim_(x→0) (((√(x+1))−1)/( ((x+1))^(1/k) −1))

Iflimx→1=xk−1x−1=L≠0Findlimx→0x+1−1x+1k−1

Answered by iloveisrael last updated on 22/May/21

Given lim_(x→1) (((x)^(1/(k )) −1)/(x−1)) = L equivalent to lim_(t→1) ((t−1)/(t^k −1)) = L ⇔lim_(x→0) (((√(x+1))−1)/( ((x+1))^(1/(k )) −1)) let x+1 = t^(2k) ; x→0 ∧ t→1 lim_(t→1) ((t^k −1)/(t^2 −1)) = (1/(lim_(t→1) (((t−1)/(t^k −1))).lim_(t→1) (t+1))) = (1/(2L))

Givenlimx→1xk−1x−1=Lequivalenttolimt→1t−1tk−1=L⇔limx→0x+1−1x+1k−1letx+1=t2k;x→0∧t→1limt→1tk−1t2−1=1limt→1(t−1tk−1).limt→1(t+1)=12L

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