1- calculate : lim ( x approaches 0 from the right (+)) x^(-lnx) >>>>>>>>>>>>>>>

Question

1- calculate : lim ( x approaches 0 from the right (+)) x^(-lnx)

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2- prove the second law of exponents ? ( if x and y are real #'s and a,b > 0 , then prove a^(x-y)=( a^x)/(a^y)

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3- show that lim (n approaches infinity )(1+(x/n))^n=e^x for any x > 0 ??

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4- Use the method to prove ...     2 arcsin (x) = arccos (1-2x^2)    x > or equal 0 ...... the methods are : integral (1/(1-x^2))dx=arcsinx+c and the other one is : integral (1/x^2+1)dx= arctanx+c

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Explanation / Answer

1)

e^ln(x^lnx) is the same as x^lnx, but I put aside the e for the end of the equation.
ln(x^lnx)= lnx*lnx.

To apply L'Hospitals Rule I have to have it in quotient form, so I rearranged my equation to lnx/ (1/lnx).

After taking the derivative of both I ended up at -(lnx)^2.

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