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The value of lim(xto2) (sqrt(1+sqrt(2+x)...

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

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)
  1. If f(x)=(2)/(x-3),g(x)=(x-3)/(x+4)," and "h(x)=-(2(2x+1))/(x^(2)+x-12)...

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  2. The value of lim(xto pi) (1+cos^(3)x)/(sin^(2)x)" is "

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

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  4. The value of lim(xto2) (2^(x)+2^(3-x)-6)/(sqrt(2^(-x))-2^(1-x))" is "

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  5. The value of lim(xto2) (((x^(3)-4x)/(x^(3)-8))^(-1)-((x+sqrt(2x))/(x-2...

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  6. If lim(xto-2^(-)) (ae^(1//|x+2|)-1)/(2-e^(1//|x+2|))=lim(xto-2^(+)) si...

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  7. lim(xto1) ((1-x)(1-x^(2))...(1-x^(2n)))/({(1-x)(1-x^(2))...(1-x^(n))}^...

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  8. The value of lim(xto(1)/(sqrt(2))) (x-cos(sin^(-1)x))/(1-tan(sin^(-1)x...

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  9. Among (i) lim(xtooo) sec^(-1)((x)/(sinx))" and "(ii) lim(xtooo) sec^(-...

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  10. lim(xtooo) ((x^(3))/(3x^(2)-4)-(x^(2))/(3x+2))" is equal to "

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  11. lim(ntooo) (n(2n+1)^(2))/((n+2)(n^(2)+3n-1))" is equal to "

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  12. lim(xtooo) ((2x+1)^(40)(4x+1)^(5))/((2x+3)^(45)) is equal to

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

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  14. lim(xtooo) (2+2x+sin2x)/((2x+sin2x)e^(sinx)) is equal to

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  15. lim(xtooo) ((x+1)^(10)+(x+2)^(10)+...+(x+100)^(10))/(x^(10)+10^(10)) i...

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  16. lim(xtooo) (2sqrt(x)+3root(3)(x)+4root(4)(x)+...+nroot(n)(x))/(sqrt((2...

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  17. If lim(ntooo) (n.3^(n))/(n(x-2)^(n)+n.3^(n+1)-3^(n))=1/3, then the ran...

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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(ntooo) (1)/(((3)/(pi)tan^(-1)2x)^(2n)+5). 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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