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The integer n for which lim(xto0) ((cosx...

The integer n for which `lim_(xto0) ((cosx-1)(cosx-e^(x)))/(x^(n))` is a finite nonzero number is__________.

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Verified by Experts

The correct Answer is:
`(3)`
`L= underset(xto0)lim((cosx-1)(cosx-e^(x)))/((1+cosx)x^(n))`
`=underset(xto0)lim(((sinx)/(x))^(2)((1-cosx)/(x)+(e^(x)-1)/(x)))/(x^(n-3))(1)/(1+cosx)`
If L is finite nonzero, then `n=3` (as for `n=1,2,L=0,` and for `n=4,L=oo`.)
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CENGAGE-LIMITS-Exercise (Numerical)
  1. If lim(xtooo) f(x) exists and is finite and nonzero and if lim(xtooo) ...

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  2. If L=lim(xto2)((10-x)^(1//3)-2)/(x-2), then the value of |1//(4L)|" is...

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  3. If lim(xto0) (p sin2x+(1-cos2x))/(x+tanx)=1, then the value of p is.

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  4. The value of lim(xtooo) ((100)/(1-x^(100))-(50)/(1-x^(50))) is .

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  5. If L= lim(xto2) (root(3)(60+x^(2))-4)/(sin(x-2)), then the value of 1/...

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  6. The value of lim(xtooo) ((20^(x)-1)/(19(5^(x))))^(1//x) is .

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  7. The value of lim(ntooo) [root(3)((n+1)^(2))-root(3)((n-1)^(2))] is .

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  8. If L=-lim(ntooo) (2xx3^(2)xx2^(3)xx3^(4)...xx2^(n-1)xx3^(n))^((1)/((n^...

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  9. The value of lim(x to oo ) (log(e)(log(e)x))/(e^(sqrt(x))) is .

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  10. If lim(xto0) (1-sqrt(cos2x).root(3)(cos3x).root(4)(cos4x)...root(n)(co...

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  11. The value of lim(x to oo ) (x-x^(2)log(e)(1+(1)/(x))) is .

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  12. Let S(n)=1+2+3+...+n " and " P(n)=(S(2))/(S(2)-1).(S(3))/(S(3)-1).(S(4...

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  13. If lim(xto1)(asin(x-1)+bcos(x-1)+4)/(x^(2)-1)=-2, then |a+b| is.

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  14. Let lim(xto1) (x^(a)-ax+a-1)/((x-1)^(2))=f(a). Then the value of f(4) ...

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  15. Number of integral values of k for which lim(xto1) sin^(-1)((k)/(log...

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  16. If lim(xto1) (1+ax+bx^(2))^((e)/((x-1)))=e^(3), then the value of bc i...

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  17. Let f''(x) be continuous at x=0 If lim(xto0) (2f(x)-3af(2x)+bf(8x))/...

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  18. If L=lim(xto0) (e^(-x^(2)//2)-cosx)/(x^(3)sinx), then the value of 1//...

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  19. The integer n for which lim(xto0) ((cosx-1)(cosx-e^(x)))/(x^(n)) is a ...

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  20. If lim(xto0) [1+x+(f(x))/(x)]^(1//x)=e^(3), then the value of ln(lim(x...

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