lim x→8 {[1+(1+x) 1/2 ] 1/2 -2}/x-8 Pls answer. sorry for this uncomfortable font


lim x→8  {[1+(1+x)1/2 ]1/2  -2}/x-8


Pls answer. sorry for this uncomfortable font



3 Answers

Ashwin Muralidharan IIT Madras
290 Points
12 years ago

Hi Saideep.


Multiply Nr and Dr by the conjugate of the Nr Twice.


When you do it for the first time, the Nr will be (1+x)1/2-3.

And the next time when you do it, the Nr will become x-8.

And this will get cancelled with the x-8, in the Dr, and then you can put x=8, in the resulting expression.


The resulting expression would be 1/{[1+(1+x)1/2 ]1/2 + 2}*{(1+x)1/2 +3}.


Put x=8 in this expression and you will get the limit as, 1/24.


Hope that helps.


All the best.


Ashwin (IIT Madras).

vbhootna vb
14 Points
12 years ago

You can directly differnsiate nr and dr sepratly twice and get answer

Bestin Johnson
31 Points
12 years ago

lim     {{1+(1+x)^1/2}^1/2}/x-8 upon direct substituition results in 0/0 form.Therefore by appying L-Hospital's rule,the limit


reduces to lim  d/dx [1/{{1+(1+x)^1/2}^1/2} / x-8]


=lim 1 / [2{{1+(1+x)^1/2}^1/2} * 1  / [2{(1+x)^1/2}


Upon putting the value of x=8,we get the required limit=1/24.

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