Intereting Posts

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I found this problem interesting to be here:

If $a,b>0$ then find the following limit: $$\lim_{x\to\pm\infty}\left(\frac{a^{\frac{1}{x}}+b^{\frac{1}{x}}}{2}\right)^x$$

Thanks!

- On $_2F_1(\tfrac13,\tfrac23;\tfrac56;\tfrac{27}{32}) = \tfrac85$ and $_2F_1(\tfrac14,\tfrac34;\tfrac78;\tfrac{48}{49}) = \tfrac{\sqrt7}3(1+\sqrt2)$
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- Why isn't the derivative of $e^x$ equal to $xe^{(x-1)}$?

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One can use L’Hospital’s Rule mechanically. The logarithm of our expression is

$$\frac{\log\left(\frac{a^{1/x}+b^{1/x}}{2}\right)}{1/x}.$$

Differentiate top and bottom. We get

$$\frac{1}{\frac{a^{1/x}+b^{1/x}}{2}}(-1/x^2)\frac{\frac{a^{1/x}\log a+b^{1/x}\log b}{2}}{(-1/x^2)}$$

Cancel the $(-1/x^2)$. The limit is $\dfrac{\log a+\log b}{2}$. Then exponentiate.

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