Questions tagged [exponential-function]
For question involving exponential functions and questions on exponential growth or decay.
8,059 questions
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How to solve $ \lim\limits_{x\to+\infty} \!\!\left(\! \frac{x^{2}+3}{3x^{2}+1}\! \right)^{\!x^{2}}\!\!\!=0\;?$
I know it can be solved with Squeeze's theorem, but I want to verify that this more conventional method might also be valid.
$$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\...
- 23
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Show that $f$ is strictly increasing on $\mathbb{R}_+.$
Problem statement:
Let $f:\mathbb{R} \to \mathbb{R}$ be defined by
$$
f(x) = a_1^x + a_2^x + \dots + a_n^x,
$$
where $n \in \mathbb{N}, \quad n \ge 3,$ and $a_1, a_2, \dots, a_n > 0,$ all of them ...
- 539
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2
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One exponent for all numbers [closed]
We can say for given $a,b,c$ value,all are real numbers
$a^b=a^c$
Since base is equal, we can conclude
$b=c$
,but why this not valid for $a=1$,what is the intuitive idea?
-4
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4
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Find $ \lim_{x\to+\infty} \left( \frac{x^{2}+3}{3x^{2}+1} \right)^{x^{2}}=0 $ [closed]
Problem
$$ \lim_{x\to+\infty} \left( \frac{x^{2}+3}{3x^{2}+1} \right)^{x^{2}}=0 $$
My Work
$$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\cdot\frac{1}{3} \right)^{x^{2}\cdot\...
1
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2
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The closure of the set of triples $(x,y,z)$ admitting non-negative integer exponents with $x^{n_1}+y^{n_2}=z^{n_3}$
I am considering the set $S'$, which extends the original problem to allow non-negative integer exponents:
$$
S'=\{(x,y,z)\in \mathbb{R}^3:\exists n_1,n_2,n_3\in\mathbb{N}_{\ge 0},\ x^{n_1}+y^{n_2}=z^{...
- 6,087
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Find $ \lim_{x\to \infty} \left( \frac{x-4}{x+1} \right)^{x+3}=e^{-5} $
$$ \lim_{x\to \infty} \left( \frac{x-4}{x+1} \right)^{x+3}=e^{-5} $$
I know that I am not making any change in the expression, I am just re-expressing it
$$ \lim_{x\to \infty} \left( 1+\frac{-5}{x+1} \...
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5
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Exponential function of negative argument [closed]
Famous $e^x$ function obeys the following well-known property:
$$
e^{-x} = \frac{1}{e^x}.
$$
I am concerned about the following. What if we didn't know that property in advance. Is it possible to ...
- 551
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Connections between the surface $x^{3}+y^{3}+z^{3}-3\cdot x\cdot y\cdot z=1$ and the curve $\left(x,y,z\right)=\left(h_{3,n}\left(t\right)\right)$?
What are other relationships between the surface $x^{3}+y^{3}+z^{3}-3\cdot x\cdot y\cdot z=1$ and (the curve defined by) functions $x=h_{3,0}\left(t\right)$, $y=h_{3,1}\left(t\right)$, $z=h_{3,2}\left(...
- 1,573
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Is there any closed form for the solution of the equation $\exp(x) - x = 2$ [closed]
I randomly gave this as a problem to myself and it seems to not be solvable by basic equation manipulation. Is this possible to solve?
- 1
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Standard names of functions defined by generalizing the power series of trigonometric and hyperbolic functions?
For integers $m$, $n$ such that $m>n\ge0$, define functions
$$
\begin{align}
h_{m,n}\left(z\right) &= \sum_{k=0}^{\infty}{\frac{z^{m\cdot k+n}}{\left(m\cdot k+n\right)!}} \\
g_{m,n}\left(z\...
- 1,573
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Explain why the function $\exp(z^2)$ has an antiderivative on the whole complex plane. [closed]
I'm working through an undergraduate course in complex analysis and am currently on a chapter entitled Cauchys integral theorem, where i encountered this question as an exercise. It doesn't seem like ...
- 11
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Conditions for the intersection of two weird functions
(From a question at HNUE High School for the Gifted)
Let $$f(x) = |a|^{bx}-|b|^{ax}\\g(x) = abx$$Such that $a$ and $b$ are real numbers.
What are the conditions needed for $f(x)$ and $g(x)$ to ...
- 11
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1
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Find $ \lim_{x\to\infty} \frac{3^{x}}{e^{x}}=+\infty$ [closed]
I see the proccedure of:
How I find the limit of $\frac{2^n}{e^{p(n)l}}$
I didn’t understand how it applies in my case:
$$ \lim_{x\to\infty} \frac{3^{x}}{e^{x}}=+\infty$$
$$ \lim_{x\to\infty} \frac{\...
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1
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71
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$x^2$ is absolutely monotonic but does not satisfy Bernstein's theorem (unless...)
I've been working for a while on something related to absolutely monotonic functions, and I've come to this realization and it feels like a significant mistake in the literature that I cannot believe ...
0
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1
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Found a strange differential equation with quaternions and $\exp()$, how do I solve?
Found this problem on a blackboard today. It looks like some form of quaternion analysis, something to do with the MacLaurin series of $e^x$ My question is, what all is going on here, and how might I ...
