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The value of underset(xto2)lim(sqrt(1+sq...

The value of `underset(xto2)lim(sqrt(1+sqrt(2+x))-sqrt(3))/(x-2)" is "`

A

`(1)/(8sqrt(3))`

B

`(1)/(sqrt(3))`

C

`8(sqrt(3))`

D

`sqrt(3)`

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
`lim _(xto2) sqrt(1+sqrt(2+x)-sqrt(3))/(x-2)`
`= lim_(xto 2) (1+sqrt(2+x-3))/((x-2){sqrt(1+sqrt(2+x))+sqrt(3)}}`
`lim_(x to 2) (sqrt(2+x)-2)/((x-2){sqrt(1+sqrt(2+x))+sqrt3})`
`=lim_(x to 2) (x-2)/((x-2){sqrt(1+sqrt(2+x))+sqrt(3)}{sqrt(2+x+2)})`.
`lim_(x to 2) (1)/({sqrt(1+sqrt(2+x))+sqrt(3)}{sqrt(2+x)+2})=(1)/(2sqrt(3)xx4)=(1)/(xsqrt(3))`
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OBJECTIVE RD SHARMA ENGLISH-LIMITS-Chapter Test
  1. The value of underset(xto2)lim(sqrt(1+sqrt(2+x))-sqrt(3))/(x-2)" is "

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  2. Let f(x)={(x^(2),x epsilonZ),((d(x^(2)-4))/(2-x),x !inZ):} the set of ...

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  3. If Sn=sum(k=1)^n ak and lim(n->oo)an=a , then lim(n->oo)(S(n+1)-Sn)/sq...

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  4. If a1=1a n da(n+1)=(4+3an)/(3+2an),ngeq1,a n dif("lim")(nvecoo)an=a , ...

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  5. If x(1)=3 and x(n+1)=sqrt(2+x(n))" ",nge1, then underset(ntooo)limx(n)...

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  6. The value of underset(xrarr0)(lim)(sqrt(1-cosx^(2)))/(1-cos x) is

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  7. Evaluate underset(ntooo)limncos((pi)/(4n))sin((pi)/(4n)).

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  8. Evaluate ("lim")(n→oo){cos(x/2)cos(x/4)cos(x/8)... cos(x/(2^n))}

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  9. If f(x) is the integral of (2 sin x - sin 2x )/(x ^ 3 ) , w...

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  10. Evaluate: ("lim")(xvec0)x^m(logx)^n ,m , n in Ndot

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  11. The value of lim(xrarroo) (logx)/(x^n), n gt 0, is

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  12. underset(xtoa)lim(log(x-a))/(log(e^(x)-e^(a)))

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  13. Let < an > be a sequence such that lim(x->oo)an=0. Then lim(n->oo)(a1...

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  14. If f(a)=2,f^(prime)(a)=1,g(a)=-1,g^(prime)(a)=2, then the value of ("l...

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  15. If f(9)=9,f^(prime)(9)=4,t h e n("lim")(nvecoo)(sqrt(f(x)-3))/(sqrt(x-...

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  16. A(i)=(x-a(i))/(|x-a(i)|),i=1,2,...,n," and "a(1)lta(2)lta(3)lt...lta(n...

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  17. lim(x -> oo) x^n / e^x = 0, (n is an integer) for

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  18. lim(xrarr0) (x)/(tan^-1x) is equal to

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  19. If f(x) =x , x<0 and f(x)=1 , x = 0, and f(x)=x^2,x>0 then lim(x->0) ...

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  20. Evaluate the following limits : Lim(x to oo) sqrt(((x+sin x)/(x- cos...

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  21. Evaluate: ("lim")(xvecoo)(1+1/(a+b x))^(c+dx),w h e r ea , b , c ,a n ...

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