Homework Help: limits

Posted by david on Saturday, December 9, 2006 at 11:58pm.

what is the derivative of e^(3*ln(x^2))

i keep getting 6/x * e^(3*ln(x^2))

but that's not one of the multiple choice so am i doing something wront.

also, what is the limit of (1-cosx)/(2*(sinx)^2) as x approaches zero. is the answer zero or nonexistent

The derivative of e^(3*ln(x^2)):
Let's write ln (x^2) as 2 ln x. Then
d/dx [e^(6 ln x)] = [e^(6 ln x)] * (6/x)

The limit of (1-cosx)/(2*(sinx)^2) as x approaches zero. The answer is NOT zero or undetermined.

Use L'Hopital's rule. The ratio of derivatives of numerator and denominator is
sin x/[4 sin x *cos x] = 1/(4 cos x)
At x = 0, that becomes 1/4

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