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lim-x-x-x-x-x-1-




Question Number 213109 by mathlove last updated on 30/Oct/24
lim_(x→∞) ((√(x+(√(x+(√x)))))/( (√(x+1))))=?
limxx+x+xx+1=?
Answered by MrGaster last updated on 30/Oct/24
=((√(x(1+(1/( (√x)))+(1/x))))/( (√(x(1+(1/x))))))  =(((√x)(√(1+(1/( (√x)))+(1/x))))/( (√x)(√(1+(1/x)))))  =((√(1+(1/( (√x)))+(1/x)))/( (√(1+(1/x)))))  lim_(x→∞) ((√(1+(1/( (√x)))+(1/x)))/( (√(1+(1/x)))))  =((√(1+0+0))/( (√(1+0))))  =((√1)/( (√1)))  =1
=x(1+1x+1x)x(1+1x)=x1+1x+1xx1+1x=1+1x+1x1+1xlimx1+1x+1x1+1x=1+0+01+0=11=1
Answered by efronzo1 last updated on 30/Oct/24
 = (√(lim_(x→∞)  ((x(1+(√((1/x)+(√(1/x^3 ))))))/(x(1+(1/x))))))   = (√( (1/1))) = 1
=limxx(1+1x+1x3)x(1+1x)=11=1
Answered by Frix last updated on 31/Oct/24
lim_(x→∞)  ((√(x+(√(x+(√x)))))/( (√(x+1)))) =lim_(t→0^+ )  ((√(1+(√(t+(√t^3 )))))/( (√(t+1)))) =1
limxx+x+xx+1=limt0+1+t+t3t+1=1

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