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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


 

Grade:11

3 Answers

Ashwin Muralidharan IIT Madras
290 Points
9 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.

Regards,

Ashwin (IIT Madras).

vbhootna vb
14 Points
9 years ago

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

Bestin Johnson
31 Points
9 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

x->8

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

               x->8

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

x->8

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

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