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

The value of `lim_(ntooo)(e^(n))/((1+(1)/(n))^(n^(2)))`is

A

-1

B

0

C

1

D

`oo`

Text Solution

Verified by Experts

The correct Answer is:
C
`L=underset(ntooo)lim(e^(n))/((1+(1)/(n))^(n^(2)))`
`=underset(ntooo)lim(e^(n))/(e^(n^(2)log_(e)(1+(1)/(n))))`
`=underset(ntooo)lime^(n-n^(2)log_(e)(1+(1)/(n)))`
Put `n=(1)/(y)`
`:.L=underset(yto0)lime^((y-log_(e)(1+y))/(y^(2)))`
`=underset(yto0)lime^((y-(y-(y^(2))/(2)+...))/(y^(2)))`
`=e^(1//2)`
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CENGAGE PUBLICATION-LIMITS-Exercises (Single Correct Answer Type)
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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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