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lim(xto0) (x^(a)sin^(b)x)/(sin(x^(c))), ...

`lim_(xto0) (x^(a)sin^(b)x)/(sin(x^(c)))`, where `a,b,c inR~{0},` exists and has non-zero value. Then,

A

0

B

`pi//2`

C

`pi`

D

`2pi`

Text Solution

Verified by Experts

The correct Answer is:
C
`underset(xto0)lim(x^(a)sin^(b)x)/(sinx^(c))=underset(xto0)limx^(a)((sinx)/(x))^(b)((x^(c))/(sinx^(c)))x^(b-c)=underset(xto0)limx^(a+b-c)`
This limit will have non-zero value of `a+b=c.`
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CENGAGE-LIMITS-Exercise (Single)
  1. lim(xto1) (xsin(x-[x]))/(x-1), where [.] denotes the greatest integer ...

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  2. lim(xtooo) (x^(2)"tan"(1)/(x))/(sqrt(8x^(2)+7x+1)) is equal to

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  3. lim(xto0) (x^(a)sin^(b)x)/(sin(x^(c))), where a,b,c inR~{0}, exists an...

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  4. lim(xto0) (x^(4)(cot^(4)x-cot^(2)x+1))/((tan^(4)x-tan^(2)x+1)) is equa...

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  5. lim(xto1) (1-x^(2))/(sin2pix) is equal to

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  6. lim(xto0) (1)/(x)cos^(1)((1-x^(2))/(1+x^2)) is equal to

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  7. lim(yto0) ((x+y)sec(x+y)-xsecx)/(y) is equal to

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  8. lim(xto1) (1+sinpi((3x)/(1+x^(2))))/(1+cospix) is equal to

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  9. lim(ntooo) sum(x=1)^(20)cos^(2n)(x-10) is equal to

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  10. lim(xto-1) ((x^(4)+x^(2)+x+1)/(x^(2)-x+1))^((1-cos(x+1))/(x+1)^(2)) is...

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  11. lim(xtooo) {(x+5)tan^(-1)(x+5)-(x+1)tan^(-1)(x+1)} is equal to

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  12. The value of lim(xto0) ([(100x)/(sinx)]+[(99sinx)/(x)]) (where [.] rep...

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  13. The value of lim(xtoa) sqrt(a^(2)-x^(2))"cot"(pi)/(2)sqrt((a-x)/(a+x))...

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  14. lim(xto0) [min(y^(2)-4y+11)(sinx)/(x)] (where [.] denotes the greatest...

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  15. The value of lim(xto0) (1-(cosx)sqrt(cos2x))/(x^(2)) is

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  16. lim(xtooo) (1)/(x+1)tan((pix+1)/(2x+2)) is equal to

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  17. The value of lim(xto1^(-)) (1-sqrt(x))/((cos^(-1)x)^(2)) is

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  18. lim(xto pi//2) (sin(xcosx))/(cos(xsinx)) is equal to

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  19. lim(xto0) [(1-e^(x))(sinx)/(|x|)] is (where [.] represents the greates...

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  20. Evaluate lim(xto0) (x(e^(x)-1))/(1-cosx) is equal to

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