Questions tagged [exponential-function]
For question involving exponential functions and questions on exponential growth or decay.
8,059 questions
-2 votes
5 answers
309 views
Solving $1^x=2$ implies $0=2$. What went wrong? [closed]
Suppose we have the exponential equation: $$1^x = 2$$ We take the natural log on both sides: $$x \cdot \ln(1) = 2$$ Which simplifies to: $$0x = 2$$ And then $$0 = 2$$ What went wrong? Is this result ...
- 424
2 votes
3 answers
207 views
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. $$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\...
- 23
1 vote
2 answers
146 views
Show that $f$ is strictly increasing on $\mathbb{R}_+.$
Problem statement: Let $f:\mathbb{R} \to \mathbb{R}$ be defined by $$ f(x) = a_1^x + a_2^x + \dots + a_n^x, $$ where $n \in \mathbb{N}, \quad n \ge 3,$ and $a_1, a_2, \dots, a_n > 0,$ all of them ...
- 405
0 votes
2 answers
101 views
One exponent for all numbers [closed]
We can say for given $a,b,c$ value,all are real numbers $a^b=a^c$ Since base is equal, we can conclude $b=c$ ,but why this not valid for $a=1$,what is the intuitive idea?
-4 votes
4 answers
252 views
Find $ \lim_{x\to+\infty} \left( \frac{x^{2}+3}{3x^{2}+1} \right)^{x^{2}}=0 $ [closed]
Problem $$ \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 vote
2 answers
67 views
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: $$ S'=\{(x,y,z)\in \mathbb{R}^3:\exists n_1,n_2,n_3\in\mathbb{N}_{\ge 0},\ x^{n_1}+y^{n_2}=z^{...
- 6,333
-1 votes
3 answers
123 views
Find $ \lim_{x\to \infty} \left( \frac{x-4}{x+1} \right)^{x+3}=e^{-5} $
$$ \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} \...
-2 votes
5 answers
186 views
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
0 votes
0 answers
47 views
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(...
- 1,843
-1 votes
2 answers
79 views
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
3 votes
1 answer
115 views
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 $$ \begin{align} h_{m,n}\left(z\right) &= \sum_{k=0}^{\infty}{\frac{z^{m\cdot k+n}}{\left(m\cdot k+n\right)!}} \\ g_{m,n}\left(z\...
- 1,843
1 vote
0 answers
86 views
Explain why the function $\exp(z^2)$ has an antiderivative on the whole complex plane. [closed]
I'm working through an undergraduate course in complex analysis and am currently on a chapter entitled Cauchys integral theorem, where i encountered this question as an exercise. It doesn't seem like ...
- 11
0 votes
0 answers
61 views
Conditions for the intersection of two weird functions
(From a question at HNUE High School for the Gifted) Let $$f(x) = |a|^{bx}-|b|^{ax}\\g(x) = abx$$Such that $a$ and $b$ are real numbers. What are the conditions needed for $f(x)$ and $g(x)$ to ...
- 11
-3 votes
1 answer
73 views
Find $ \lim_{x\to\infty} \frac{3^{x}}{e^{x}}=+\infty$ [closed]
I see the proccedure of: How I find the limit of $\frac{2^n}{e^{p(n)l}}$ I didn’t understand how it applies in my case: $$ \lim_{x\to\infty} \frac{3^{x}}{e^{x}}=+\infty$$ $$ \lim_{x\to\infty} \frac{\...
0 votes
1 answer
71 views
$x^2$ 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 ...
