...not as long as Mathstud's. So feel free to create a new one

Some (if not all) are easy, so @ the masters (not http://www.mathhelpforum.com/math-he...s/masters.html), pardon me for insulting your abilities and please lend normal people 10 more minutes than you need for solving them

Of course, $\displaystyle \color{red}\huge \text{NO L'HOSPITAL'S RULE}$

(you may be able to use the gradient definition). No power series is needed but approximations could be needed in some limits.

I think solutions can all be less than 5 or 6 lines...

1/

$\displaystyle \lim_{x \to \tfrac \pi 2} \quad \left(\frac \pi x -1\right)^{\tan(x)}$

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2/

$\displaystyle \lim_{x \to \tfrac \pi 2} \quad 4x \tan(2x)-\frac{\pi}{\cos(2x)}$

Solution #1 (Chris L T521) & book's answer

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3/

$\displaystyle \lim_{x \to 0} \quad \frac{(x+27)^{\tfrac 13}-3}{(x+16)^{\tfrac 14}-2}$

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4/

$\displaystyle \lim_{x \to 0} \quad \left(\frac{a^x+b^x}{2}\right)^{\tfrac 1x}$

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5/

$\displaystyle \lim_{x \to 0} \quad \frac{e^x-e^{\sin(x)}}{x-\sin(x)}$

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6/

$\displaystyle \lim_{x \to 0} \quad \frac{x^{\sqrt{x}}}{(\sqrt{x})^x}$

Solution #1 (Mathstud)

Alternative

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7/

$\displaystyle \lim_{x \to 1} \quad \frac{2}{1-x^2}-\frac{3}{1-x^3}$

Solution #1 (Mathstud)

Alternative

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8/

$\displaystyle \lim_{x \to 1} \quad \frac{\ln(\sin(\tfrac \pi 2 x))}{(x-1)^2}$

Solution #1 (Mathstud)

Alternative

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9/

$\displaystyle \lim_{x \to 1} \quad \frac{1-x^\alpha}{\ln(x)}$

Solution #1 (Mathstud)

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10/

$\displaystyle \lim_{x \to \tfrac \pi 6} \quad \frac{(x-\tfrac \pi 6)^2}{2 \sin(x)-1}$

Solution #1 (Mathstud)

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11/

$\displaystyle \lim_{x \to a} \quad \frac{a-x-a \ln(a)+a \ln(x)}{a-(2ax-x^2)^{\tfrac 12}}$

Have fun