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Evaluate the following limits: lim(xra...

Evaluate the following limits:
`lim_(xrarr0)((1-cos2x))/((cos2x-cos8x))`

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Verified by Experts

The correct Answer is:
`1/15`
Given limit `=lim_(xto0)(2sin^(2)x)/(2sin5xsin3x)=lim_(xto0)(sinx.sinx)/(sin5x.sin3x)`
`=(lim_(xto0)((sinx)/(x)).lim_(xto0)((sinx)/(x)))/(lim_(xto0)((sin5x)/(5x)xx5)xxlim_(xto0)((sin3x)/(3x)xx3))`
`=(1xx1)/((1xx5)xx(1xx3))=1/15.`
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RS AGGARWAL-LIMIT-EXERCISE 27B
  1. Evaluate the following limits: lim(xrarr0)((sin3x+sin5x))/((sin6x-si...

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  2. Evaluate the following limits: lim(xrarr0)([sin(2+x)-sin(2-x)])/(x)

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  3. Evaluate the following limits: lim(xrarr0)((1-cos2x))/((cos2x-cos8x)...

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

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  5. Evaluate the following limits: lim(xrarr0)((sqrt(1+2x)-sqrt(1-2x)))/...

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  6. Evaluate the following limits: lim(xrarr0)((e^(tan-x)-1))/(tanx)

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  7. Evaluate the following limits: lim(xrarr0)((e^(3+x)-sinx-e^(3)))/(x)

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  8. Evaluate the following limits: lim(xrarr0)((e^(tanx)-1))/(tanx)

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  9. Evaluate the following limits: lim(xrarr0)((e^(tanx)-1))/(x)

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  10. Evaluate the following limits: lim(xrarr0)(ax+x cosx)/(b sinx)

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  11. Evaluate the following limits: lim(xrarr0)(sinax+bx)/(ax+sinbx),wher...

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  12. lim(x->pi)(sin(pi-x)/(pi(pi-x)))

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  13. lim(x->pi/2) (tan 2x)/(x-pi/2)

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  14. Evaluate the following limits: lim(xrarr0)(cos2x-1)/(cosx-1)

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  15. Evaluate the following limits: lim(xrarr0)(cosec x-cotx)

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  16. Evaluate the following limits: lim(xrarr0)(1-cos2mx)/(1-cos2nx)

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

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  18. Evaluate the following limits: lim(xrarr0)(sin^(2)mx)/(sin^(2)nx)

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  19. Evaluate the following limits: lim(xrarr0)(sin2x+sin3x)/(2x+sin3x)

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  20. The value of lim(x->0)(sec4x-sec2x)/(sec3x-secx) is

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