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If L=lim(xto0) (e^(-x^(2)//2)-cosx)/(x^(...

If `L=lim_(xto0) (e^(-x^(2)//2)-cosx)/(x^(3)sinx)`, then find the value of `1//(3L)` is ________.

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Verified by Experts

The correct Answer is:
`(4)`
`underset(xto0)lim(e^(-x^(2)//2)-cosx)/(x^(3)sinx)`
`=underset(xto0)lim((1-((x^(2)//2))/(1!)+((x^(2)//2)^(2))/(2!))-(1-x^(2)/(2!)+x^(4)/(4!)))/(x^(3)(x-(x^(3))/(3!)))`
`=underset(xto0)lim(((x^(4))/(8))-((x^(4))/(24)))/(x^(4)(1-(x^(2))/(3!)))=(1)/(12)`
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CENGAGE PUBLICATION-LIMITS-Numerical Value Type
  1. The value of lim(xtooo) ((100)/(1-x^(100))-(50)/(1-x^(50))) is .

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

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

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

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

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

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  7. Find k if x = 3 is a root of equation kx^2 – 10x + 3 = 0

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

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  9. Let Sn=1+2+3++n and Pn=(S2)/(S2-1)dot(S3)/(S3-1)dot(S4)/(S4-1)...(Sn)...

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

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

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

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  13. If lim(xto1) (1+ax+bx^(2))^((c)/((x-1)))=e^(3), then find the value of...

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

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  15. If L=lim(xto0) (e^(-x^(2)//2)-cosx)/(x^(3)sinx), then find the value o...

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  16. The integer n for which lim(x rarr0)((cos x-1)(cosx-e^x))/x^n is a fin...

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

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  18. The largest value of the non-negative integer a for which lim(xrarr1){...

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  19. Find the 24th term of the sequence: 12, 10, 8, 6,……

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  20. Let alpha,betainR be such that lim(xto0) (x^(2)sin(betax))/(alphax-sin...

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