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If lim xto0(x^(-3)sin3x+a x^(-2)+b) exis...

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

A

`a=-3`

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 PUBLICATION-LIMITS-Exercises (Single Correct Answer Type)
  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)(n x^(n+1)-(n+1)x^n+1)/((e^x-e)sinpix),\ where \n=100 , ...

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  4. lim(x->0)(log(1+x+x^2)+"log"(1-x+x^2))/(secx-cosx)=

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  5. The value of lim(xtooo) (root(3)(x^(3)+2x^(2))-sqrt(x^(2)+x)) is

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  6. The value of lim(x->0)(1+sinx-cosx+"log"(1-x))/(x^3) is (a)1/2 ...

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  7. If lim(xtoa)f(x)=1 and lim(xtoa)g(x)=oo then lim(xtoa){f(x)}^(g(x))=e^...

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

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

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  10. ("lim")(x -> 0)((1+tanx)/(1+sinx))^(cos e cx) is equal to (a)e ...

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  11. The value of lim(x->1)(2-x)^(tan((pix)/2)) is (a)e^(-2/pi) (b) e^...

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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(ntooo) {((n)/(n+1))^(alpha)+"sin"(1)/(n)}^(n) (where alphainQ) is ...

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

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  16. The value of lim(xrarr0)(1^x+2^x+3^x+...+n^x)^(a//x)/n, is:

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  17. The value of ("lim")(xvec1)(p/(1-x^p)-q/(1-x^q)),p ,q , in N , equal ...

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

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  19. lim(x->oo)cot^(-1)(x^(-a)loga x)/(sec^(-1)(a^xlogx a)),(a >1)is equal ...

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

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