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Evaluate: lim(xto0)((x-1+cosx)/(x))^(1...

Evaluate:
`lim_(xto0)((x-1+cosx)/(x))^(1//x)`

Text Solution

Verified by Experts

The correct Answer is:
`(1)/(sqrte)`
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CHHAYA PUBLICATION-MISCELLANEOUS EXAMPLES-CALCULUS
  1. Evaluate: lim(xto(pi)/(2))(1+cosx)^(3secx)

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  2. Evaluate: lim(xto(1)/(2))(sin((pix)/(2))-cos((pix)/(2)))/((1)/(2)-x)

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  3. Evaluate: lim(xto0)((x-1+cosx)/(x))^(1//x)

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  4. Evaluate: lim(xto2)(sqrt(1+sqrt(2+x))-sqrt3)/(x-2)

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  5. Evaluate: lim(xto0)(sinx+cosx)^(1//x)

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  6. Evaluate: lim(xto0)[tan(x+(pi)/(4))]^(1//x)

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  7. Evaluate: lim(xto0)(1)/(x^(3))[sin(3x+a)-3sin(2x+a)+3sin(x+a)-sina]

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  8. Evaluate: lim(xto2)(sqrt(2x+5)-3(sqrt(2x-3)))/(root3(2x-3))

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  9. Evaluate: lim(xto0)(1+sin2x)^(cosecx)

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  10. Evaluate: lim(xto1)(x^(x)-1)/(xlogx)

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  11. Evaluate: lim(xto0)(log(1+x+x^(2))+log(1-x+x^(2)))/(secx-cosx)

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  12. Prove that lim(xto0)(xsqrt(2ax-x^(2)))/((sqrt(8ax-4x^(2))+sqrt(8ax))^(...

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  13. Evaluate [without using L' Hospital's rule]: lim(xto2)(2-sqrt(2+x))/...

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  14. Evaluate [without using L' Hospital's rule]: lim(xto(pi)/(4))(4sqrt2...

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  15. Find the differential coefficient of: cos(ax^(2)+bx+c)

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  16. Find the differential coefficient of: tan^(2)(ax)

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  17. Find the differential coefficient of: x^(x)

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  18. Find from first principle the differential coefficient of: xtan^(-1)...

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  19. Examine the differentiability of f(x)=|sinx-cosx|" at "x=(pi)/(4)

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  20. Prove that the function f(x)=|x| is not differentiable there at x=0.

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