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Evaluate the following limits: lim(xra...

Evaluate the following limits:
`lim_(xrarr0)((1-cos2x))/((1-cos6x))`

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Verified by Experts

The correct Answer is:
`4/9`
Given limit `=lim_(xto0)(2sin^(2)2x)/(esin^(2)3x)=lim_(xto0)(sin^(2)2x)/(sin^(2)3x)`
`=lim_(xto0)(((sin2x)/(2x))^(2)xx4x^(2))/(((sin3x)/(3x))^(2)xx9x^(2))=4/9.(lim_(2xto0)((sin2x)/(2x))^(2))/(lim_(3xto0)((sin3x)/(3x))^(2))`
`=((4)/(9)xx(1)/(1))=4/9.`
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RS AGGARWAL-LIMIT-EXERCISE 27B
  1. Evaluate the following limits: lim(xrarr0)((1-cos2x))/(sin^(2)2x)

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  2. Evaluate the following limits: lim((1-cos2x))/(3tan^(2)x)

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  3. Evaluate the following limits: lim(xrarr0)((1-cos2x))/((1-cos6x))

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  4. Evaluate, underset(xrarr0)"lim"(1-cosmx)/(1-cosnx)

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  5. Evaluate underset(xrarr0)"lim"(2sinx-sin2x)/(x^(3))

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  6. Evaluate the following limits: lim(xrarr0)((tanx-sinx))/(sin^(3)x)

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  7. Evaluate the following limits: lim(xrarr0)((tan2x-sin2x))/(x^(3))

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  8. lim(xrarr0)("cosec"x-cotx)/(x) is equal to

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  9. Evaluate the following limits: lim(xrarr0)((cot2x-cosecex))/(x)

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  10. Evaluate the following limits: lim(xrarr0)((cosecx-cotx))/(x^(3))

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  11. lim(xrarr(pi//4))(sec^(x)-2)/(tanx-1) is

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  12. Evaluate the following limits: lim(xrarr(pi)/(4))((cosec^(2)x-2))/((...

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  13. Evaluate the following limits: lim(xrarr (pi)/(4))(tanx-1)/((x-(pi)/(...

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  14. lim(x->pi)((sin3x-3sinx)/((pi-x)^3))

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  15. Evaluate: ("lim")(xvecpi/2)(1+cos2x)/((pi-2x)^2)

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  16. Evaluate the following limits: lim(xrarra)((sinx-sina))/((x-a))

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  17. Evaluate the following limits: lim(xrarra)((sinx-sina))/((x-a))

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  18. Evaluate the following limits: lim(xrarra)((sinx-sina))/((sqrtx-sqrta...

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  19. Evaluate the following limits: lim(xrarr0)((sin5x-sin3x))/(sinx)

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  20. Evaluate the following limits: lim(xrarr0)((cos3x-cos5x))/(x^(2))

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