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("lim")(xto0)((1+tanx)/(1+sinx))^(cos e ...

`("lim")_(xto0)((1+tanx)/(1+sinx))^(cos e cx)i se q u a l t o` (a)e (b) `1/e` (c) 1 (d) none of these

A

`underset(xto0)limf(x)` exists for `ngt0`

B

`underset(xto0)limf(x)` does not exists for `nlt0`

C

`underset(xto0)limf(x)` does not exists for any value of n

D

`underset(xto0)limf(x)` exists for any value of n

Text Solution

Verified by Experts

The correct Answer is:
C
`underset(xto0)lim((1+tanx)/(1+sinx))^(cosecx)=underset(xto0)lim((1+tanx)^((1)/(sinx)))/((1+sinx)^((1)/(sinx)))`
`=(underset(xto0)lim((1+tanx)^((1)/(tanx)))^((1)/(cosx)))/((1+sinx)^((1)/(sinx)))`
`=(e^((1)/(cos0)))/(e)=1`
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CENGAGE-LIMITS-Exercise (Single)
  1. ("lim")(xto0)((2^m+x)^(1/m)-(2^n+x)^(1/n))/x is equal t o (a) 2(1/(m2...

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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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