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Prove that: underset(xrarr0)lim ([1+3x]^...

Prove that: `underset(xrarr0)lim ([1+3x]^((x+3)/x))= e^9`

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UNITED BOOK HOUSE-MISCELLANEOUS EXERCISE-EXERCISE
  1. Prove that: underset(xrarrinfty)lim ((x+5)/(x+1))^x = e^4

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  2. Prove that: underset(xrarrpi/2)lim (1+cosx)^(3secx) = e^3

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  3. Prove that: underset(xrarr0)lim ([1+3x]^((x+3)/x))= e^9

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  4. Evaluate the following limits: underset(x rarr0)lim(xe^x - log(x+1))/(...

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  5. Evaluate the following limits: underset(x rarr0)lim[tan(pi/4+x)]^(1/x)

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  6. Evaluate the following limits: underset(x rarr0)lim(sinx + cos x)^(1/x...

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  7. Evaluate the following limits: underset(x rarr0)limsinlog(1+x)/log(sin...

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  8. It g(x) is the inverse of f(x) and f(X) = (1+x^3)^-1, show that g'(x) ...

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  9. Find the differential coefficients of the folloiwng functions: x^x

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  10. Find form 1st principle the differential coefficients of the folloiwng...

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  11. Find form 1st principle the differential coefficients of the folloiwng...

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  12. Find dy/dx when x^(siny) + y ^(sinx) = 1

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  13. Find dy/dx when x^(logx) + (sinx)^x + 15 x

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  14. Find (d^2y)/(dx^2) when x= e^-t and y = te^-1

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  15. Find (d^2y)/(dx^2) when x= a sin^3t and y = a cos^3 t at t = pi/4

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  16. Find (d^2y)/(dx^2) when x=a(t-sint) and y = a (1+cost) at t = pi/...

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  17. If y^2 (1-x^2) = x^2 +1, show that (1-x^4) (dy/dx)^2 = y^4 -1.

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  18. If x = sec theta - cos theta, y = sec^n theta - cos^n theta, then show...

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  19. If y= x^(y^x), prove that y1 = (ylogy(1+x logx logy)/(x log x (1-xlogy...

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  20. or,find the derivatives of tan^(-1) ((sqrt(1+x^2)-1)/x) with respect t...

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