For question involving exponential functions and questions on exponential growth or decay.

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8
votes
6answers
2k views

About $\lim \left(1+\frac {x}{n}\right)^n$

I was wondering if it is possible to get a link to a rigorous proof that $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$
11
votes
5answers
2k views

Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$

I was messing around with the definition of the derivative, trying to work out the formulas for the common functions using limits. I hit a roadblock, however, while trying to find the derivative of ...
10
votes
5answers
885 views

Solve the equation $2^x=1-x$

Solve the equation: $$2^x=1-x$$ I know this is extremely easy and I know the solution using graphical approach. Basically, I can see the solution, but I can't work it out algebraically.
37
votes
19answers
3k views

Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
10
votes
3answers
1k views

Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series

In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following: $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over ...
19
votes
18answers
2k views

How to understand why $x^0 = 1$, where $x$ is any real number?

Alright, so the idea of an exponent, $x$, is that you are multiplying its base by itself $x$ number of times. With base $5$ and $x=3$, we have that $5^3$ = $5 \cdot 5 \cdot 5$ I understand that the ...
8
votes
2answers
533 views

Integration by parts: $\int e^{ax}\cos(bx)\,dx$

I need to evaluate the following function and then check my answer by taking the derivative: $$\int e^{ax}\cos(bx)\,dx$$ where $a$ is any real number and $b$ is any positive real number. I know ...
4
votes
2answers
224 views

How to solve this equation $x^{2}=2^{x}$?

How to solve this equation $$x^{2}=2^{x}$$ where $x \in \mathbb{R}$. Por tentativa erro consegui descobri que $2$ é uma solução, mas não encontrei um método pra isso. Alguma sugestão?(*) ...
21
votes
9answers
1k views

Finding the limit of $\left(\frac{n}{n+1}\right)^n$

Find the limit of: $$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$ I'm pretty sure it goes to zero since $(n+1)^n > n^n$ but when I input large numbers it goes to $0.36$. Also, when ...
12
votes
3answers
519 views

Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?

In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one? I can of course ...
6
votes
3answers
834 views

Finding asymptotes of exponential function and one-sided limit

Find the asymptotes of $$ \lim_{x \to \infty}x\cdot\exp\left(\dfrac{2}{x}\right)+1. $$ How is it done?
7
votes
7answers
2k views

How would you explain why “e” is important? (And when it applies?) [duplicate]

Possible Duplicate: Intuitive Understanding of the constant “e” Let's say you want to explain this to your teenage son. I understand the technical definition of $e$ $$ ...
21
votes
13answers
2k views

Intuitive proofs that $\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n=e^x$

At this link someone asked how to prove rigorously that $$ \lim_{n\to\infty}\left(1+\frac xn\right)^n = e^x. $$ What good intuitive arguments exist for this statement? Later ...
19
votes
3answers
372 views

What combinatorial quantity the tetration of two natural numbers represents?

Tetration is a generalization of exponentiation in arithmetic and a part of a series of other generalized notions, Hyperoperators. Consider $m\uparrow n$ denotes the tetration of $m$ and $n$. i.e. ...
3
votes
1answer
240 views

How can one prove the impossibility of writing $ \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions?

Can we express $ \displaystyle \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions? (Note: Infinite series are not allowed.) If not, then is there a proof that $ \displaystyle \int ...
3
votes
1answer
189 views

Can the graph of $x^x$ have a real-valued plot below zero?

The function $f(x) = x^x$ gives a complex number only if x has an even denominator. I'm not sure about irrational numbers. Why, then, is the best graph I can find of that function that of Wolfram ...
2
votes
1answer
82 views

The exponential extension of $\mathbb{Q}$ is a proper subset of $\mathbb{C}$?

This question come from a recent post Exponential extension of $\mathbb{Q}$. An exponential field is a field $\mathbb{K}$ where it's well defined a function $E:\mathbb{K} \rightarrow \mathbb{K}$ ...
2
votes
2answers
135 views

Exponential extension of $\mathbb{Q}$

A non-trivial exponential function $E:\mathbb{K} \rightarrow \mathbb{K}$ in a field $\mathbb{K}$ is a function such that \begin{split} E(x+y)=E(x)E(y) \quad \forall x,y \in \mathbb{K} \\ E(x)=1 \iff ...
41
votes
4answers
2k views

If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?

It is known that if two matrices $A,B \in M_n(\mathbb{C})$ commute, then $e^A$ and $e^B$ commute. Is the converse true? If $e^A$ and $e^B$ commute, do $A$ and $B$ commute? Edit: Addionally, what ...
42
votes
11answers
24k views

Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?
4
votes
2answers
435 views

Closed form for $n$-th derivative of exponential

I need the closed-form for the $n$-th derivative ($n\geq0 $): $$\frac{\partial^n}{\partial x^n}\exp\left(-\frac{\pi^2a^2}{x}\right)$$ Thanks! By following the suggestion of Hermite polynomials: ...
7
votes
3answers
469 views

How was $e$ first calculated?

I understand how $\pi$ is calculated, but I am interested in references that explain when and how the natural exponent $e$ was developed. What mathematical principles are behind the value of $e$?
9
votes
7answers
799 views

How to prove continuity of $e^x$.

I simply want a proof that $e^x$ is continuous. I have never really been able to find something satisfying these points: $e$ is defined to be the limit $\lim_{n\to\infty}\left(1+{1\over ...
6
votes
3answers
240 views

Prove that $e^{-A} = (e^{A})^{-1}$

Let $A, B \in R^{n \times n}$. Prove that $e^{-A} = (e^{A})^{-1}$. ($R$ is the real numbers) I've tried messing around with both sides, evaluated as sums. I just can't get the two to match up. Any ...
6
votes
10answers
3k views

Proving $\lim \limits_{n\to +\infty } \left(1+\frac{x}{n}\right)^n=\text{e}^x$.

I knew that $e^x=\lim \limits_{n\to+\infty }{\left(1+\frac{x}{n}\right)^n}$. But I've never seen its proof. So I tried to prove it using $\exp(\ln x)=\ln(\exp(x))=x$. Here is what I've tried so far : ...
4
votes
8answers
1k views

Prove that $e^x\ge x+1$ for all real $x$ [duplicate]

Without using differentiation, logarithmic function, rigorously, prove that $$e^x\ge x+1$$ for all real values of $x$.
3
votes
3answers
219 views

Is it trivial to say $\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k}$

Is it trivial to say $$\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k},$$ considering the fact that we know $$\mathop {\lim }\limits_{n \to \infty } {(1 + {1 \over n})^n} = e?$$ ...
14
votes
4answers
423 views

Euler's identity: why is the $e$ in $e^{ix}$? What if it were some other constant like $2^{ix}$?

$e^{ix}$ describes a unit circle in polar coordinates on the complex plane, where x is the angle (in radians) counterclockwise of the positive real axis. My intuition behind this is that ...
10
votes
2answers
2k views

Show that $ e^{A+B}=e^A e^B$

If $A$ and $B$ are $n\times n$ matrices such that $AB = BA$ (that is, $A$ and $B$ commute), show that $$ e^{A+B}=e^A e^B$$ Note that $A$ and $B$ do NOT have to be diagonalizable.
11
votes
1answer
152 views

Reason for LCM of all numbers from 1 .. n equals roughly $e^n$

I computed the LCM for all natural numbers from 1 up to a limit $n$ and plotted the result over $n$. Due to the fast-raising numbers, I plotted the logarithm of the result and was surprised to find a ...
19
votes
4answers
4k views

“What if” math joke: the derivative of $\ln(x)^e$

Randall Munroe, the creator of xkcd in his latest book What if writes (p. 175) that the mathematical analog of the phrase "knock me over with a feather" is seeing the expression $ \ln( x )^{e}$. And ...
4
votes
2answers
3k views

Why is $e^{x}$ not uniformly continuous on $\mathbb{R}$?

It seems intuitively very clear that $e^{x}$ is not uniformly continuous on $\mathbb{R}$. I'm looking to 'prove' it using $\epsilon$-$\delta$ analysis though. I reason as follows: Suppose $\epsilon ...
2
votes
5answers
870 views

How does $e^{i x}$ produce rotation around the imaginary unit circle?

Euler’s formula states that $e^{i x} = \cos(x) + i \sin(x)$. I can see from the MacLaurin Expansion that this is indeed true; however, I don’t intuitively understand how raising $e$ to the power of ...
1
vote
0answers
57 views

Find the positive root of the equation $ce^{-c}-2(1-e^{-c})^2=0$

Can you help me find a root for $c$ in the equation below? $$ce^{-c}-{10\over5}(1-e^{-c})^2=0$$ By expanding this I got, $$ce^{-c}-2 + 4 e^{-c}-2e^{-2c}=0$$ now grouping, ...
13
votes
6answers
461 views

$\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]

If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
5
votes
3answers
108 views

Stuck on this integral involving exp and the floor function

Here is the integral $$\int_0^\infty \lfloor x \rfloor e^{-x}dx$$ Here is what I have so far: $$I = \sum_{n=0}^\infty \int_n^{n+1} n e^{-x}dx$$ $$ = \sum_{n=0}^\infty -ne^{-n-1} + ne^{-n}$$ $$ = ...
3
votes
2answers
244 views

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 ...
3
votes
1answer
223 views

integral evaluation of an exponential

let be the function $$ e^{-a|x|^{b}} $$ with $ a,b $ positive numbers bigger than zero then how could i evaluate this 2 integrals ? $$ \int_{-\infty}^{\infty}dxe^{-a|x|^{b}}e^{cx}$$ here 'c' can ...
10
votes
4answers
914 views

Confused about complex numbers

I am confused about something: \begin{eqnarray} (e^{2 i \pi})^{0.5} = (e^{2 i \pi \cdot 0.5})= e^{i \pi}=-1 \end{eqnarray} but \begin{eqnarray} e^{2 i \pi}=1~ and~ 1^{0.5}=1 \end{eqnarray} ...
8
votes
3answers
315 views

Proving that a definition of e is unique

We can define $e$ as the number such that $\lim_{h \to 0} \frac{e^h-1}{h}=1$. However, of course we can only define $e$ this way if it is unique, i.e., there is no other value $c$ for which that is a ...
6
votes
3answers
151 views

How to prove $(\frac{n+1}{e})^n<n!<e(\frac{n+1}{e})^{n+1}$ without integrating method?

How to prove $$\left(\frac{n+1}{e}\right)^n<n!<e\left(\frac{n+1}{e}\right)^{n+1}$$ without integrating method? In fact we could prove this by noticing that $$i<x<i+1\Rightarrow \ln ...
4
votes
3answers
221 views

Prove that if $\phi'(x) = \phi(x)$ and $\phi(0)=0$, then $\phi(x)\equiv 0$. Use this to prove the identity $e^{a+b} = e^a e^b$.

I am given the following. hint Consider $f(x)=e^{-x} \phi(x)$. I am unsure how to approach this problem.
4
votes
3answers
256 views

Iterates of $f_b(x) = x - \log_b(x) $ - for $\log(b) \approx 0.399$: convergence to accumulation points or chaos?

In a previous question I arrived at the family of functions (depending on a real parameter b for the base of exponentiation/logarithm): $$ f(x) = x - \log_b(x) \qquad \qquad b \gt 0$$ with the ...
3
votes
3answers
759 views

Limit involving $(\sin x) /x -\cos x $ and $(e^{2x}-1)/(2x)$, without l'Hôpital

Find: $$\lim_{x\to 0}\ \frac{\dfrac{\sin x}{x} - \cos x}{2x \left(\dfrac{e^{2x} - 1}{2x} - 1 \right)}$$ I have factorized it in this manner in an attempt to use the formulae. I have tried to use that ...
2
votes
7answers
197 views

How to estimate the value of $e$. [closed]

I am currently studying how to estimate $e$. To solve this problem I use these methods discuss below: Method 1: We know that $e^x = 1 + \dfrac{x}{1!} + \dfrac{x}{2!}+ \cdots $ So if we consider a ...
1
vote
6answers
124 views

How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$

How to prove $$1+x \leq e^x~\forall x \in \mathbb{R}$$ I'm stuck, I tried taking logs but didn't know how to proceed.
1
vote
1answer
2k views

How would I create a exponential ramp function from 0,0 to 1,1 with a single value to explain curvature?

I need an exponential function that will take linear input from 0,0 to 1,1 and give me back an exponential shaped curve such that changes in X near the 0 point result in small increases in Y, but each ...
6
votes
5answers
1k views

How to integrate $\int \frac{e^x dx}{1\,+\,e^{2x}}$

Ok, I give up, I have tried with $u$-substitution and integration by parts but I can't solve it. The integral is: $$\int{\frac{e^x dx}{1+e^{2x}}}$$ I have tried $u=e^x$, $u=e^{2x}$ and also ...
3
votes
3answers
656 views

Exponential function and uniform convergence of polynomials.

How can I prove that no sequence of polynomials converges uniformly to the exponential function? Thanks in advance for any help.
3
votes
2answers
130 views

Inequality $(1+\frac1k)^k \leq 3$

How can I elegantly show that: $(1 + \frac{1}{k})^k \leq 3$ For instance I could use the fact that this is an increasing function and then take $\lim_{ k\to \infty}$ and say that it equals $e$ and ...