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### Evaluate $\lim_{x\to 0} \frac{a^x -1}{x}$ without applying L'Hopital's Rule. [duplicate]

The questions is: Evaluate $$\lim_{x\to 0} \frac{a^x -1}{x}$$ without applying L'Hopital's Rule. Does this question fundamentally same as asking if the $\lim_{x\to 0} \frac{a^x -1}{x}$ exists? rather ...
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### Elementary proof that $2^x$ is derivable [duplicate]

I was wondering if there was an elementary proof, so not using the exponential function, that $2^x$ is derivable. I define the function $f(x) = 2^x$ by $f(a/b) = \sqrt[b]{2^a}$ for a and b integers, ...
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### How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my math ...
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### Proving $(1 + 1/n)^{n+1} \gt e$

I'm trying to prove that $$\left(1 + \frac{1}{n}\right)^{n+1} > e$$ It seems that the definition of $e$ is going to be important here but I can't work out what to do with the limit in the ...