L'Hospital's Rule for tan(x) and tan(10x) at 9pi/4: Exact Value Calculation

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lim (x approaches 9pi/4) (tanx)^tan10x

use L'Hospital's Rule to find the exact valuemy attempt:
y = lim (tanx)^tan10x
lny= lim ln(tanx)^tan10x
= tan10x lim ln(tanx)

I really don't know where to go from there

-Thanks
 
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In your last line, you take tan10x out of the limit without evaluating it. Put it back in and turn it into cot(10x) to put it on the bottom, then apply L'Hospital's. Make sure to apply exp at the end to get the final answer.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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