# Questions tagged [exponential-function]

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

524 questions
13answers
52k views

### How to prove that exponential grows faster than polynomial?

In other words, how to prove: For all real constants $a$ and $b$ such that $a > 1$, $$\lim_{n\rightarrow\infty}\frac{n^b}{a^n} = 0$$ I know the definition of limit but I feel that it's not ...
11answers
10k views

### Comparing $\pi^e$ and $e^\pi$ without calculating them

How can I calculate, without calculator or similar device, the values of $\pi^e$ and $e^\pi$ in order to compare them?
9answers
15k views

### Proof that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$

I was wondering the following. And I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function ...
9answers
14k 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 $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$
24answers
14k 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 ...
2answers
2k views

4answers
975 views

### Why $\lim\limits_{n\to \infty}\left(1+\frac{1}{n}\right)^n$ doesn't evaluate to 1?

I am trying to identify what the flaw is exactly when reasoning about a limit such as the definition of $\mathbf e$: $$\lim_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^n={e}$$ Now, I know ...
2answers
621 views

3answers
475 views

### How to find $\lim_{n \to \infty}|\left( 1+\frac{z}{n}\right)^{n}|$? [duplicate]

This is a step on the way to proving $$\lim_{n \to \infty}\left(1 + \frac{z}{n}\right)^{n} = e^{z}.$$ I'm looking for an answer without 1) a summation or 2) a logarithmic function. So far, I ...
4answers
2k views

### Proving the irrationality of $e^n$

Let $n$ be a positive integer. I know the traditional proof that $e$ is irrational. How do we show that $e^n$ is irrational in some sort of similar line? I am of course assuming it is but I would be ...
1answer
604 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 ...
2answers
1k 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 that ...
5answers
2k 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.
5answers
6k views

4answers
448 views

### How to prove $\lim\limits_{n \to \infty} (1+\frac1n)^n = e$?

How to prove the following limit? $$\lim_{n \to \infty} (1+1/n)^n = e$$ I can only observe that the limit should be a very large number! Thanks.
6answers
24k views

### A question comparing $\pi^e$ to $e^\pi$ [duplicate]

I was doing an algebra problem set following a chapter on logarithms and exponentiation, and it presented this "bonus question": Without using your calculator, determine which is larger: $e^\pi$ or ...
7answers
39k views

### Why does the sum of the reciprocals of factorials converge to $e$?

I've been asked by some schoolmates why we have $$\sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with ...
8answers
33k 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 $x$.
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 ...
5answers
719 views