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If lim(xto0) (x^(-3)sin3x+ax^(-2)+b) exi...

If `lim_(xto0) (x^(-3)sin3x+ax^(-2)+b)` exists and is equal to 0, then

A

`a=1`

B

`a=0`

C

`b=1`

D

`b=-1`

Text Solution

Verified by Experts

The correct Answer is:
A
`underset(xto0)lim((sin3x)/(x^(3))+(a)/(x^(2))+b)=underset(xto0)lim(sin3x+ax+bx^(3))/(x^(3))`
`=underset(xto0)lim(3(sin3x)/(3x)+a+bx^(2))/(x^(2))`
For existence,
`(3+a)=0`
or `a=-3`
`:. L=underset(xto0)lim(sin3x-3x+bx^(3))/(x^(3))`
`=27underset(t to0)lim(sint-t)/(t^(3))+b=0(3x=t)`
`=-(27)/(6)+b=0`
or `b=(9)/(2)`
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CENGAGE-LIMITS-Exercise (Single)
  1. lim(xto0) ((2^(m)+x)^(1//m)-(2^(n)+x)^(1//n))/(x) is equal to

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

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  3. lim(xto1) (nx^(n-1)-(n+1)x^(n)+1)/((e^(x)-e)sinpix), where n=100,is eq...

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  4. lim(xto0) (log(1+x+x^(2))+log(1-x+x^(2)))/(secx-cosx)=

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  5. The value of lim(xto0) (root(3)(x^(3)+2x^(2))-sqrt(x^(2)+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. lim(xto0) (cos(tanx)-cosx)/(x^(4)) is equal to

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  8. If lim(xto0) (x^(-3)sin3x+ax^(-2)+b) exists and is equal to 0, then

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  9. If lim(xto0) (x^(n)sin^(n)x)/(x^(n)-sin^(n)x) is non-zero finite, then...

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  10. lim(xto0) ((1+tanx)/(1+sinx))^(cosecx) is equal to

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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(mtooo) ("cos"(x)/(m))^(m) is

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  13. lim(ntooo) ((n^(2)-n+1)/(n^(2)-n-1))^(n(n-1)) is equal to

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  14. lim(ntooo) {((n)/(n+1))^(a)+"sin"(1)/(n)}^(n) (where alphainQ) is equa...

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  15. lim(xtooo) [((e)/(1-e))((1)/(e)-(x)/(1+x))]^(x) is

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  16. lim(x->0)((1^x+2^x+3^x+....+n^x)/n)^(1/x)

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  17. The value of lim(x to 1) ((p)/(1-x^(p))-(q)/(1-xq)),p,q,inN, equals

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  18. lim(xtooo) (x(logx)^(3))/(1+x+x^(2)) equals

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  19. lim(xtooo) (cot^(-1)(x^(-a)log(a)x))/(sec^(-1)(a^(x)log(x)a)),(agt1), ...

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

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