# Tagged Questions

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### Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...
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### Why can we first take the limit that goes to e?

For example \begin{aligned} \lim_{n \to \infty} \left(1 + \frac{1}{ \frac{n-1}{2}} \right)^{n} &= \lim_{n \to \infty} \left(1 + \frac{1}{ \frac{n-1}{2}} ...
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### $\frac{db^x}{dx}$ without $e$

For no other reason other than interest, I'm trying to find the general derivative of $b^x$ without using a definition of $e$ from a different context. I feel like, chronologically in history, this ...
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### Is the function $f(x)=1^x=1$ considered an exponential function?

I am confused about the following: The exponential function (by definition) is a function of the form $f(x)=a^x$ where $a>0$. However, when $a=1$, we get the constant function $f(x)=1^x=1$. Is the ...
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### What $\alpha$ such that if $xy=\alpha$, then $e^{-x}+e^{-y}\geq 2e^{-\sqrt \alpha}$?

For every $x,y \gt 0$, if $xy=\alpha$, then we have $$e^{-x}+e^{-y}\geq 2e^{-\sqrt \alpha}$$ What are the possible values of $\alpha$? $2 < e^{1/(n+1)} + e^{-1/n}$ led to this problem. ...
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### Yacov Perelman Nepero game

This is my first question, so sorry if I'll make any mistake in using the site formatting. I found this game on a book by Yacov Perelman and I thought it could be nice to introduce Nepero number to ...
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### Proving properties about complex exponential

I defined $a^z$ for $z \in \mathbb{C}$ as $a^z = \exp(z\log(a))$ and I proved it is continous, now I want to show that $a^n = a \cdot a \cdot a \cdot \ldots \cdot a$ for $n \in \mathbb{N}$ so ...
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### Continuity and other properties of complex exponential

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces ...
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### Prove logarithm rules using definition as the inverse exponential

Problem $3$. Show that $\operatorname{exp} : \Bbb R \to (0, \infty)$ is bijective. Its inverse function is called the (natural) logarithm $\log : (0, \infty) \to \Bbb R$. Verify the logarithm ...
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### Proving that the exponential function is bijective

Prove that $\exp: \mathbb{R} \mapsto (0,\infty)$ is a bijection. Okay, so the first part is really easy: injectivity follows directly from writing the exponential function as a series. ...
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### $0^0$ — indeterminate, or $1$? [duplicate]

One of my teachers argued today that 0^0 = 1. However, WolframAlpha, intuition(?) and various other sources say otherwise... 0^0 doesn't really "mean" anything.. can anyone clear this up with some ...
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### Minimizing bizarre exponential function

I am trying to minimize the function in the form of $f(x) = (1-a^x)^x$ where $0 < a < 1$ with respect to $x$ (for $x > 0$) and I am stuck! Unfortunately the derivative is not nice enough to ...
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### Solve equation $\tfrac 1x (e^x-1) = \alpha$

I have the equation $\tfrac 1x (e^x-1) = \alpha$ for an positive $\alpha \in \mathbb{R}^+$ which I want to solve for $x\in \mathbb R$ (most of all I am interested in the solution $x > 0$ for ...
Suppose that $f:\mathbb{R}\rightarrow (0,\infty)$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ are two functions and satisfies the following relation $$f(xy)=f(y)^{g(x)}$$ ...