Recently, I've been messing around with factorials (mainly because I was just told that \(x! \approx (\frac{x}{e})^x\times\sqrt{2\pi x}\)). I came across a couple calculus problems involving the function. Since I am very new to calculus, I figured I should ask the community on ** how** to get the answers and what they are, not just by typing them into Wolfram Alpha and seeing what comes out.

Since \(x!\) increases so rapidly, I decided taking the \(x\) root of \(x!\). This gave me the function, \(\color{Red}{f(x)=x!^{\frac{1}{x}}}\). That function is the red graph in the picture below.

I noticed that, even though \(f(0)\) is undefined (since\(\frac{1}{0}\) is undefined), it still appears to have a value. That is my first problem: to find

\[\lim_{x\rightarrow 0} \color{Red}{f(x)}\]

Next, I noticed that \(\color{red}{f(x)}\) is **not** linear. However, it appeared to be linear, so I decided to graph the derivative, which gave me the function \(\color{Blue}{g(x)=\dfrac{\text{d}}{\text{d}x}~f(x)}\). This is the blue graph in the picture above. Since \(\color{red}{f(x)}\) approached linearity (?), I knew that \(\color{blue}{g(x)}\) must have a limit as it approached infinity. That leads me to my second problem: to find

\[\lim_{x\rightarrow \infty} \color{blue}{g(x)}\]

The graph is at this link.

*NOTE: The domains of both functions stop before 171 because 171! is a massive number, to the point that most online calculators can't handle.*

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## Comments

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TopNewestAccording to Wolfram Alpha, \( \lim_{x\rightarrow 0 } f(x) \) is an interesting value.

It is not immediately apparant to me why this is true as yet.

Note: \(\gamma\) is the Euler-Mascheroni constant.

Note that \( f(x) \approx \frac{x}{e} \) for larget values. Hence, if there is any justice in the world (meaning that if the limit exists), it is most likely that \( \lim g(x) = \frac{1}[e} \).

However, there is a slight flaw in your logic. Namely, the following statement is not true: "If \( | f(x) - g(x) | < \epsilon \) for all \(x\), then \( \lim g'(x) = \lim f'(x) \).".

An extra condition will need to be added to arrive at "If \( f(x) \approx g(x) \) (in some manner), then \( \lim g'(x) = \lim f'(x) \)."

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I did some experimenting with factorials, here are some interested facts:

Those limits are derived from the gamma function, and the second one seem to evaluate to \(\frac{1}{e}\)

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I updated the comment

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