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If f(x)=lim(nrarroo) (cos(x)/(sqrtn))^(n...

If `f(x)=lim_(nrarroo) (cos(x)/(sqrtn))^(n)`, then the value of `lim_(xrarr0) (f(x)-1)/(x)` is

A

0

B

1

C

2

D

`3//2`

Text Solution

Verified by Experts

The correct Answer is:
A
`f(x)=underset(nrarroo)(lim)(cos.(x)/(sqrtn))^(n)`
`" "=e^(underset(nrarroo)(lim)(cos.(x)/(sqrtn)-1).n)`
`" "e^(-(2underset(nrarroo)(lim)(x^(2))/(4)(sin^(2).(x)/(2sqrtn))/((x^(2))/(4sqrtn))))`
`" "=e^(-(1)/(2)x^(2))`
`rArr" "underset(xrarr0)(lim)(f(x)-1)/(x)=underset(xrarr0)(lim)(e^(-(1)/(2)x^(2))-1)/(x)=0`
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CENGAGE-LIMITS-Single Correct Answer Type
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  2. If agt0, b gt0 than lim(nrarroo) ((a-1+b^((1)/(n)))/(a))^(n)=

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  3. If f(x)=lim(nrarroo) (cos(x)/(sqrtn))^(n), then the value of lim(xrarr...

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  4. lim(xrarr0) (log(x^(x^(2))+2sqrtx))/(tansqrtx) is equal to

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  5. Let f:RrarrR be such that f(a)=1, f(a)=2. Then lim(x to 0)((f^(2)(a+x)...

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  6. The value of lim(nrarroo) ((n^(2)-n+1)/(n^(2)-n-1))^(n(n-1) is

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  7. If f(n)=("lim")(xvec0)"{"(1+s in x/2)(1+s in x/(x^2))(1+s in x/(x^n))"...

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  8. lim(nrarroo) (1-x+x.root n e)^(n) is equal to

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  9. underset(xrarr1)(lim)(root(13)x-root7x)/(root5x-root3x) is

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  10. lim(xrarr1) (root(13)x-root7x)/(root5x-root3x) is

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  11. The value of lim(x rarr 0) (1-cos2x)/(e^(x^(2))-e^(x)+x) is

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  12. If f(a)=(1)/(4), then lim(hrarr0) (f(a+2h^(2))-f(a-2h^(2)))/(f(a+h^(3)...

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  13. (lim)(xvec0^+)1/(xsqrt(x))("a t a n"^(-1)(sqrt(x))/a-btan^(-1)(sqrt(x)...

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  14. The value of lim(xrarr0) ((1+2x)/(1+3x))^((1)/(x^(2))).e^((1)/(e^(x)))...

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  15. If F:R rarr R be a differentiable function at x = 0 satisfying f(0)=0 ...

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  16. The value of lim(xrarr(3pi)/(4)) (1+root3(tanx))/(1-2cos^(2)x) is

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  17. Let g(x)=((x-1)^(n))/(logcos^(m)(x-1)),0ltxlt2 m and n integers, m ne0...

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  18. Number of integral values of lambda for which lim(xrarr1) sec^(-1)((la...

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  19. If lim(xrarr0) (e^(ax)-e^(x)-x)/(x^(2))=b (finite), then

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  20. If lim(xrarr0)(x^3)/(sqrt(a+x)(bx-sinx))=1,agt0, then a+b is equal to

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