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

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  3. The value of lim(ntooo) [1/n+2] is

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  4. ("lim")(xvec0)(log(1+x+x^2)+"log"(1-x+x^2))/(secx-cosx)= -1 (b) 1 ...

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  5. The value of lim(xto0) (2^(3x)-3^(2x))/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. The value of lim(ntooo) [(1+2+3+...+n)/n^2] is

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  8. If ("lim")(xvec0)(x^(-3)sin3x+a x^(-2)+b) exists and is equal to 0, th...

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  9. If lim(x->0)(x^n-sinx^n)/(x-sin^n x) is non-zero finite, then n must b...

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  10. ("lim")(xto0)((1+tanx)/(1+sinx))^(cos e cx)i se q u a l t o (a)e (b)...

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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")(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")(nvecoo)"{"(n/(n+1))^(alpha)+sin(1/n)]^n(when alpha in Q) is e...

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  15. ("lim")(xvecoo)[(e/(1-e))(1/e-x/(1+x))]^xi s e^((1-e)) (b) e^(((1-e)...

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  16. ("lim")(xvec0)((1^x+2x+3^x++n^x)/n)^(1//x)i se q u a lto (n !)^n (b)...

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  17. Ifp > q >0a n dp r<-1<q r , then find the value of tan^(-1)(p-q)/(1+p ...

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  18. ("lim")(xvecoo)(x(log"x")^3)/(1+x+x^2)e q u a l 0 (b) -1 (c) 1 ...

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  19. ("lim")(xvecoo)(cot^(-1)(x^(-a)(log)a x)/(sec^(-1)(a^x(log)x a),(a >1)...

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

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