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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-LIMITS-Exercise (Single)
  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) (nx^(n-1)-(n+1)x^(n)+1)/((e^(x)-e)sinpix), where n=100,is eq...

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  4. lim(xto0) (log(1+x+x^(2))+log(1-x+x^(2)))/(secx-cosx)=

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  5. The value of lim(xto0) (root(3)(x^(3)+2x^(2))-sqrt(x^(2)+x)) is

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  6. The value of lim(xto0) (1+sinx-cosx+log(1-x))/(x^(3)) is

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  7. lim(xto0) (cos(tanx)-cosx)/(x^(4)) is equal to

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

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

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  10. lim(xto0) ((1+tanx)/(1+sinx))^(cosecx) is equal to

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  11. The value of lim(xto1) (2-x)^(tan((pix)/(2))) is

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  12. The value of lim(mtooo) ("cos"(x)/(m))^(m) is

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  13. lim(ntooo) ((n^(2)-n+1)/(n^(2)-n-1))^(n(n-1)) is equal to

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

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

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  16. lim(x->0)((1^x+2^x+3^x+....+n^x)/n)^(1/x)

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  17. The value of lim(x to 1) ((p)/(1-x^(p))-(q)/(1-xq)),p,q,inN, equals

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

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  19. lim(xtooo) (cot^(-1)(x^(-a)log(a)x))/(sec^(-1)(a^(x)log(x)a)),(agt1), ...

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

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