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Evaluate the following limits: lim(x t...

Evaluate the following limits:
`lim_(x to0)(sqrt2-sqrt(1+cosx))/(2x+sin3x)`

Text Solution

Verified by Experts

The correct Answer is:
`(1)/(4sqrt2)`
`lim_(xto0)(sqrt2-sqrt(1+cosx))/(sin^(2)x)=lim_(xto0)(sqrt2-sqrt(2cos^(2)(x//2)))/((1-cos^(2)x))=lim_(xto0)(sqrt2(1-cos""(x)/(2)))/((1-cosx)(1+cosx))`
`=lim_(xto0)(sqrt2(1-cos""(x)/(2)))/((2sin^(2)""(x)/2)(1+cosx))=(1)/(sqrt2)lim_(xto0)((1-cos""(x)/(2)))/((1-cos^(2)""(x)/(2))(1+cosx))`
`=(1)/(sqrt2)lim_(xto0)(1)/((1+cos""(x)/(2))(1+cosx))=(1)/(sqrt2)xx(1)/((1+1)(1+1))=(1)/(4sqrt2).`
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RS AGGARWAL-LIMIT-EXERCISE 27B
  1. lim(x->pi)(sin(pi-x)/(pi(pi-x)))

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

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

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

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

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

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

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

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

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  10. Evaluate the following limits: lim(x to0)(sqrt2-sqrt(1+cosx))/(2x+si...

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  11. Evaluate the following limits: lim(xrarr0)(sqrt(1+sinx)-1sqrt(1-sinx...

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  12. lim(x->pi/6)(2-sqrt(3)cosx-sinx)/((6x-pi)^2)

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  13. Evaluate the following limits: lim(xrarr0)(cos ax-cos bx)/(cos cx-1)

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  14. Evaluate the following limits: lim(xrarra)(cosx-cosa)/(cotx-cota)

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  15. lim(x rarr pi/4)(tan^3x-tanx)/(cos(x+pi/4)

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  16. Evaluate the following limits: lim(xrarr(pi)/(2))(sqrt2-sqrt(1+sinx)...

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  17. Evaluate the following limit: (lim)(x->pi/6)(cot^2x-3)/(cos e c x-2)

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  18. Evaluate the following limits: lim(xrarr pi)(sqrt(2+cosx)-1)/((pi-x)...

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  19. Evaluate the following limits: lim(xrarr(pi)/(4))(1-tanx)/(1-sqrt2si...

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  20. Evaluate the following limits: lim(x to(pi)/(6))(2sin^(2)x+sinx-1)/(2...

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