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lim(xtooo) [((e)/(1-e))((1)/(e)-(x)/(1+...

`lim_(xtooo) [((e)/(1-e))((1)/(e)-(x)/(1+x))]^(x)` is :

A

limit does not exist when `a=pi//6`

B

`L=-1` when `a=pi`

C

`L=1` when `a=pi//2`

D

`L=1` when `a=0`

Text Solution

Verified by Experts

The correct Answer is:
C
`underset(xtooo)lim[((e)/(1-e))((1)/(e)-(x)/(1+x))]^(x)=e^(underset(xtooo)lim[((e)/(1-e))((1)/(e)-(x)/(1+x))-1]x)`
`=e^(underset(xtooo)lim[(1)/(1-e)-(e)/(1-e)(x)/(1+x)-1]x)`
`=e^(underset(xtooo)lim[(e)/(1-e)-(e)/(1-e)(x)/(1+x)]x)`
`=e^(underset(xtooo)lim(e)/(1-e)[1-(x)/(1+x)]x)`
`=e^((e)/(1-e)underset(xtooo)lim(x)/(1+x))`
`=e^((e)/(1-e)underset(xtooo)lim(1)/(1+(1)/(x)))`
`=e^((e)/(1-e))`
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CENGAGE PUBLICATION-LIMITS-Exercises (Single Correct Answer Type)
  1. lim(xto0) ((2^(m)+x)^(1//m)-(2^(n)+x)^(1//n))/(x) is equal to

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  2. The value of lim(ntooo) [(1)/(n)+(e^(1//n))/(n)+(e^(2//n))/(n)+...+(e^...

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  3. ("lim")(xto1)(n x^(n+1)-(n+1)x^n+1)/((e^x-e)sinpix),\ where \n=100 , ...

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  4. lim(x->0)(log(1+x+x^2)+"log"(1-x+x^2))/(secx-cosx)=

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  5. The value of lim(xtooo) (root(3)(x^(3)+2x^(2))-sqrt(x^(2)+x)) is

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  6. The value of lim(x->0)(1+sinx-cosx+"log"(1-x))/(x^3) is (a)1/2 ...

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  7. If lim(xtoa)f(x)=1 and lim(xtoa)g(x)=oo then lim(xtoa){f(x)}^(g(x))=e^...

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  8. If lim xto0(x^(-3)sin3x+a x^(-2)+b) exists and is equal to 0, then

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  9. If lim(xto0)(x^n-sinx^n)/(x-sin^n x) is non-zero finite, then n must b...

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  10. ("lim")(x -> 0)((1+tanx)/(1+sinx))^(cos e cx) is equal to (a)e ...

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  11. The value of lim(x->1)(2-x)^(tan((pix)/2)) is (a)e^(-2/pi) (b) e^...

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  12. The value of ("lim")(xvecoo)(cos"x"/"m")^("m") is 1 (b) e (c) e^(-1...

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  13. ("lim")(xvecoo)((n^2)/(n^2))^(n(n-1)i se q u a lto e (b) e^2 (c) e^(...

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  14. lim(ntooo) {((n)/(n+1))^(alpha)+"sin"(1)/(n)}^(n) (where alphainQ) is ...

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  15. lim(xtooo) [((e)/(1-e))((1)/(e)-(x)/(1+x))]^(x) is :

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  16. The value of lim(xrarr0)(1^x+2^x+3^x+...+n^x)^(a//x)/n, is:

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  17. The value of ("lim")(xvec1)(p/(1-x^p)-q/(1-x^q)),p ,q , in N , equal ...

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  18. lim(xtooo) (x(logx)^(3))/(1+x+x^(2)) equals

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  19. lim(x->oo)cot^(-1)(x^(-a)loga x)/(sec^(-1)(a^xlogx a)),(a >1)is equal ...

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  20. The value of lim(ntooo)(e^(n))/((1+(1)/(n))^(n^(2)))is

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