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The value of `("lim")_(xto2)(sqrt(1+sqrt(2+x))-sqrt(3))/(x-2)i s` (a)`1/(8sqrt(3))` (b) `1/(4sqrt(3))` (c) 0 (d) none of these

A

0

B

`e^(x)`

C

`log_(e)x`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
`underset(xto2)lim(sqrt(1+sqrt(2+x))-sqrt(3))/(x-2)`
`underset(xto2)lim(1+sqrt(2+x)-3)/((sqrt(1+sqrt(2+x))+sqrt(3))(x-2))" "`(Rationalizing)
`=underset(xto2)"lim"(sqrt(2+x)-2)/((sqrt(1+sqrt(2+x))+sqrt(3)")"(x-2)))`
`=underset(xto2)"lim"((x-2))/("("sqrt(1+sqrt(2+x))+sqrt(3)")""("sqrt(2+x)+2")""("x-2")")`
(Rationalizing)
`=(1)/("("2sqrt(3")")4)=(1)/(8sqrt(3))`
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CENGAGE-LIMITS-Exercise (Single)
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  2. The value of lim(xto pi) (1+cos^(3)x)/(sin^(2)x)" is "

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  11. ("lim")(nvecoo)(n(2n+1)^2)/((n+2)(n^2+3n-1))i se q u a lto 0 (b) 2...

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  12. ("lim")(xvecoo)((2x+1)^(40)(4x-1)^5)/((2x+3)^(45))i se q u a lto 16 (...

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  13. lim(xtooo) [sqrt(x+sqrt(x+sqrt(x)))-sqrt(x)] is equal to

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  16. ("lim")(xvecoo)(2sqrt(x)+3sqrt(x)+4sqrt(x)+...+nsqrt(n))/(sqrt((2x-3))...

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  17. If("lim")(xvecoo)(n .3^n)/(n(x-2)^n+n .3^(n+1)-3^n)=1/3, t h e nt h e ...

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  18. lim(ntooo) n^(2)(x^(1//n)-x^(1//(n+1))),xgt0, is equal to

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  19. Let f(x)=("lim")(xvecoo)1/((3/pitan^(-1)2x)^(2n)+5)dot Then the set of...

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  20. f(x)=("ln"(x^(2)+e^(x)))/("ln"(x^(4)+e^(2x))). Then lim(x to oo) f(x) ...

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