Questions tagged [exponential-function]

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

530 questions
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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 ...
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How do pocket calculators calculate exponents?

I'd like to know specifically how a pocket calculator (TI calculators also apply) calculates $e^{0.1}$, and what methods or algorithms pocket calculators use in order to produce their answer.
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$\pi$ 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 ...
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Show that $e^{-\beta} = \frac{1}{\sqrt{\pi}} \int_{0}^{\infty} \frac{e^{-u}}{\sqrt{u}} e^{-\beta^2 / 4u} du$.

Show that $e^{-\beta} = \frac{1}{\sqrt{\pi}} \int_{0}^{\infty} \frac{e^{-u}}{\sqrt{u}} e^{-\beta^2 / 4u} du$. I'm not really sure of how I should proceed to show this, and it's pretty un-intuitive as ...
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Efficient Method to find funsum$(n) \pmod m$ where funsum$(n)= 0^0 + 1^1 + 2^2 + … n^n$

I have a series: $$f(n) = 0^0 + 1^1 + 2^2 + ..... n^n$$ I want to calculate $f(n) \pmod m$ where $n ≤ 10^9$ and $m ≤10^3$. I have tried the approach of the this accepted answer but the complexity is ...
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Exponential Equation with base -1, 0, and 1

I'm a high school student that learning exponential equations. In my book there is a mathematic problem about exponential equations written as following format : $$h(x)^{f(x)}=h(x)^{g(x)}$$ The ...
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Limit $\lim (1+(a^{1/n}-1) / b)^n$

How can I find the $\lim (1+(a^{1/n}-1) / b)^n$ when $n \to \infty$? I would greatly appreciate it if you kindly give me some hints.
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Why is $a=e$ the smallest number such that $a^x\ge 1+x$ for all $x$? [duplicate]

Calculus book: Find all numbers $a$ such that $\forall x, a^x \ge 1+x$ I immediately thought of the inequality $e^x\ge 1+x$ and guessed that the answer was any number $a$ in $[e,\infty)$. After ...
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Evaluating $\lim_{n \to \infty} n\left(1-\frac{1}{e}\left(1+\frac{1}{n}\right)^{n} \right)$

$$\lim_{n \to \infty} n\bigg(1-\dfrac{1}{e}\bigg(1+\dfrac{1}{n}\bigg)^{n} \bigg)$$ If I write expansion of $\bigg(1+\dfrac{1}{n}\bigg)^{n}$ it was equal to expansion of $e$ so $n-n=0$. Is limit is ...
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Proof that the series expansion for exp(1) is a Cauchy sequence

Consider the series expansion for the exponential function at x = 1: $$a_n := \sum\limits_{i = 0}^n{\frac{1}{i!}}$$ I want to prove that this is a Cauchy sequence, using the remainder formula for ...
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Find the average value of the function $f(t) = 10(e^{5t} - 1)$ over the interval $[0, 1]$.

I have the following solution for my question but I am not sure if I have done the steps right:
Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$ [closed]
I'm not sure how to go about this. Does it involve using MVT? I got as far as saying $e = \frac{e^x - e^y}{x-y}$.
$a_n=(1+\frac{1}{n})^n$ , $b_n=\sum_{k=0}^n \frac{1}{k!}$. Show that $b_n-\frac{3}{2n} < a_n < b_n$.