Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.

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Why is $\lim\limits_{x \to +\infty}\frac{x \sqrt{x+2}}{\sqrt{x+1}} - x = \frac12$?

I need to evaluate this limit: $$\lim_{x \to +\infty}\frac{x \sqrt{x+2}}{\sqrt{x+1}}-x$$ to calculate the asymptote of this function: $$\frac{x \sqrt{x+2}}{\sqrt{x+1}}$$ which, according to the class ...
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18 views

Pade approximant of infinite order

The Pade approximant states that you can approximate a function $f(x)$ by a rational function $R(x)$ of a given order. My question is, if the order of $R(x)$ goes to infinity, does $R(x)$ approach $f(...
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1answer
29 views

Is it possible to factor out monomials in a rational function?

after thinking about the problem for some hours I thought I come here to ask. My problem is that I want to do a coordinate transformation on the following equation $y=\frac{a}{x^2}+\frac{b}{x}+c+dx+...
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275 views

Partial fractions and using values not in domain

I'm studying partial fraction decomposition of rational expression. In this video the guy decompose this rational expression: $$ \frac{3x-8}{x^2-4x-5}$$ this becomes: $$\frac{3x-8}{(x-5)(x+1)} = \...
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A question about the log of a rational function

We have the rational function : $$f(x)=\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}\;\;\;,\;\;n\in \mathbb{Z}^{+}$$ It's not hard to prove that : $$\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}=(-1)^{n}\prod_{k=1}^{n-1}\...
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1answer
30 views

System of (quite similar) two equations

Given the real, natural and binary successions $\{t_1,...,t_N\} \in \mathbb{R}$, $\{n_1,...,n_N\} \in \mathbb{N}$ and $\{E_1,...,E_N\} \in \{0,1\}$ we would like to find the $(x,y)$ that satisfies the ...
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26 views

Simplfying complex rational expression

I'm trying to simplify $$ \frac {\dfrac {x}{y} - \dfrac {y}{x}}{y}.$$ My method of trying to solve this is try to simplify the numerator $\frac {x}{y}-\frac{y}{x}$ Then I find the GCD: $xy$, multiply, ...
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Indefinite integral of a rational function with linear denominator: $ \int \frac{ x^7}{(x+1)}{dx} $

$$ \int \frac{ x^7}{(x+1)}{dx} $$ $$ \int \frac{ \left(x^7 + x^6 - x^6 - x^5 + x^5 + x^4 -x^4 - x^3 + x^3 + x^2 - x^2 -x^1 + x^1 +1 -1\right ) }{\left(x+1\right)}{dx}$$ $$ \int { \left(x^6 - x^5 + x^...
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Ambiguous integral

What is the real integral of the function $$f(x) =\frac{1-x^2}{(1 + x^2)^2}$$? Is it $F_1(x) = \frac{x}{1 + x^2} + C$ or $F_2(x) = \arctan x + C$ ? The brochure I was reading gave the first result ...
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1answer
18 views

Rational bijection with rational inverse from the reals to the unit interval

I am looking for a rational bijection from the reals to the open unit interval $(0,1)$ in such a way that the inverse is also rational. The main purpose is to change the interval of integration from ...
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Entries of the inverse of $\left[\frac{1}{x+i+j-1}\right]_{i,j\in\{1,2,\ldots,n\}}$ are polynomials in $x$.

Let $n$ be a positive integer. Define $$\textbf{A}_n(x):= \left[\frac{1}{x+i+j-1}\right]_{i,j\in\{1,2,\ldots,n\}}$$ as a matrix over the field $\mathbb{Q}(x)$ of rational functions over $\mathbb{Q}$ ...
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The ordered field of rational functions does not have the LUB property, can someone give an example?

S is a set of some rational functionals and is bounded above, but S has no least upper bound. Can someone give an example of S?
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1answer
26 views

Algebraic series, rational fraction of two variables in the form of polynomial

I come across the following claim: Let $y\in\mathbb{C}[[x]]$ be an algebraic series, that is, there exist $n\in\mathbb{N}^*$ $A_i(x)\in\mathbb{C}[x]$ for $i=0,...,n$ and $A_n(x)\neq 0$ such that \...
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How to integrate $\int \frac{x^{13}\ dx}{x^5 + 1}$

We get this problem from our teacher today. I only wish that it was $x^{14}$ in the numerator, so we can use substitution method: $$\int \dfrac{x^{13}\ dx}{x^5 + 1}$$ I can't find way to integrate ...
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37 views

Degree of field extension and rational function field

Suppose that $k$ and $k'$ are fields such that $k\subset k'$ and $[k':k]=n$, where $n$ is an positive integer. Do we have $[k'(x):k(x)]=[k':k]$? Why? Thanks for your help!
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Automorphism of $\mathbb C(x,y)$ and its order in $\mathrm{Aut}(\mathbb C(x,y))$

In the following problem Let $M=\bigg (\begin{matrix} a& b\\ c& d\end{matrix}\bigg )$ be a nonsingular matrix with integer coefficients and $L=\mathbb C(x,y)$. (i) Show that $\phi(x)=...
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1answer
37 views

Plot of a function

What is the plot of: $$y=\frac{\beta(1-\alpha)x}{\alpha(1-\beta)+(\beta-\alpha)x}$$ with $0<\alpha<\beta<1$. How do I handle the parameters? How do I compute the derivatives to check for ...
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1answer
98 views

Luröth's Theorem

I've been struggling trying to understand the Jacobson's Basic Algebra vol. II proof of the Luröth's theorem. Let $K$ be a field, $K(X)$ the field of rational fonctions and take $L$ to be a sub-...
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1answer
60 views

Why do some rational functions approach the asymptote from the other side?

There are some rational functions that do a switcheroo when $x \to -\infty$ where (when graphed) they cross the horizontal asymptote in order to approach from the other side. It only seems to happen ...
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Simplify $f(x)=\Gamma(n/2)/(\Gamma(1) \Gamma(n/2-1))$… a Rational Expression using the Gamma Function.

I was reviewing a document about an algorithm wherein it is stated that $f(x)$ is a probability density function: (1)$$ f(x)=\frac{\Gamma(\frac{n}{2})}{\Gamma(1)\Gamma(\frac{n}{2}-1)}\frac{2}{n-2}\...
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Fixed Point in the Space of Rational Functions

Let $\mathcal R$ be the space of rational functions and $F: \mathcal R \to \mathcal R $ be a function that transforms a rational function into another rational function. Is there a fixed point ...
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Subtracting $\frac{(x+3)}{(x^2-1)} - \frac{(x-2)}{(x^2+2x+1)}$

$\frac{(x+3)}{(x^2-1)} - \frac{(x-2)}{(x^2+2x+1)}$ To solve the problem I first dissembled the equation on the denominator $ \frac{(x+3)}{(x-1)*(x+1)} - \frac{(x-2)}{(x+1)^2}$ I multiplied the ...
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Showing that the slope of a complicate function is greater than 1

I have a $V$ function: $$V_n(x, y) = -\frac{1+2x}{7+5x+n+2nx} + \frac{-1+2x}{2-n+x(5+2n)} - \frac{(1-2y)^2}{(2-n+y(5+2n))^2} + \frac{4(2+y)^2}{(7+n+y(5+2n))^2}.$$ Here's the input form in case you ...
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1answer
37 views

Prove the inequality involving exponential function in form of $\exp( \frac{1}{x} )$

For $\nu > 0$, $0 < x \leq \nu $, and a positive integer $S$, (we think) following an inequality always holds $1- \left( \frac{1}{x+1} \right)^S \geq \exp \left( -\frac{1}{Sx} \right) $ Does ...
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Constructing a rational function from its asymptotes

Question: Give an example of a rational function that has vertical asymptote $x=3$ now give an example of one that has vertical asymptote $x=3$ and horizontal asymptote $y=2$. Now give an example of a ...
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Why is $y=\frac{x+y}{x}$ the same as $y=\frac{x}{x-1}$?

Why is $y=\frac{x+y}{x}$ the same as $y=\frac{x}{x-1}$? It seems to be the same on graphing calculators, but I don't get why. To generalize my statement, I would like to know how to simplify ...
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1answer
22 views

Three-dimensional curve whose coordinates are rational functions

Are there three real rational functions $f,g,h$ with no poles in $[0,1]$, such that $f\geq 0,g\geq 0,f+g \geq h \geq 0$ on $[0,1]$ and the curve $\gamma(t)=(f(t),g(t),h(t)) (t\in [0,1])$ passes ...
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A function that is locally a quotient of polynomials but not globally [duplicate]

Let $X =\{ x_1x_4=x_2x_3\;, (x_2,x_4) \neq (0,0)\} \subset \mathbb{C^4}$, i.e. not both of $x_2,x_4$ are zero. Define a function $\phi$ on $X$ by $\phi(x)=\left\{\begin{matrix} \frac{x_1}{x_2} & ,...
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Approximating the normal CDF for $0 \leq x \leq 7$

In answer http://stackoverflow.com/a/23119456/2421256, an approximation of the complementary normal CDF (ie $\frac{1}{\sqrt{2 \pi}} \int_x^{+\infty} e^{-\frac{t^2}{2}} dt$) was given for $0 \leq x \...
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Why is a polynomial also a rational function?

In a recent question, I asked about non-standard-looking rational functions, i.e., something that was not in the classic numerator-denominator form. I was told that all polynomials are rational ...
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Creating a rational function with specific parameters

For a game I made, tetris, the blocks must go faster and faster every level, I want the speed to be $500$ at level $1$, and $+-250$ at level $6$ ($500$ means, $1$ block is moving down per $500$ $ms$)...
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Evaluating the rational integral $\int \frac{x^2+3}{x^6(x^2+1)}dx $

Evaluate $$\int \frac{x^2+3}{x^6(x^2+1)}dx .$$ I am unable to break into partial fractions so I don't think it is the way to go. Neither is $x=\tan \theta$ substitution. Please give some hints. ...
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Basic equation solving $t/(1+t)=1-1/(1+t)$

In found this equality in my math book, could anyone explain to me why it is equal? $t/(1+t)=1-1/(1+t)$
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1answer
31 views

Must a rational function always be in numerator-denominator form?

This may seem trivial, but I'm looking at two examples from high school math books and wondering if they are really examples of rational functions. The first is a line $\overline{DT}$ made up of two ...
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If f/g is symmetric (resp homogeneous), must f and g be as well?

Suppose we have two polynomials $f$, $g$ in $k[X_1, ..., X_n]$ over some field $k$, and they have no factor in common. Suppose that $f/g$ is symmetric. Must than $f$ and $g$ also both be symmetric? ...
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When a Rational function becomes a line?

Why is it that when $AD = BC$, this equation becomes a horizontal line? $$y = \frac {Ax+B}{Cx+D} $$ For any other values where $AD$ isn't equal to $BC$ it is a rational function.
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Cancelling variable factors in a rational function

Consider the function $\displaystyle\frac{2x−1}{x+5}$. The domain of this function is all real numbers except $x = -5$. Now consider that I do this: $\displaystyle\frac{2x−1}{x+5}⋅\frac{x}{x}$. This ...
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Why do rational functions like $\frac{1}{x}$ have weird shapes? [closed]

As many know, rational functions have weird shapes like hyperbolas placed diagonally. They are in couple parts, for example $\frac{1}{x}$. Why does that happen?
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How to show that $X^p-t\in\mathbb{F}_p(t)[x]$ is irreducible? [duplicate]

This question is previously asked here, but there is no complete solution of it. I understand that the root $\alpha$ exist in the algebraic closure of $\mathbb{F}_p(t)[x]$, and it is the only root ...
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Does every non-Archimedean ordered field contain $\mathbb Q (x)$ as an ordered subfield?

Every ordered field $\mathbb F$ contains $\mathbb Q$ in a canonical way. If the field is not Archimedean there exists an $x>n$ for all $n \in \mathbb N$. Since we are dealing with a field any ...
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1answer
35 views

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$

Simplify $\frac{5x}{x^2 - x - 6} + \frac{4}{x^2 + 4x + 4}$. How come the answer is left as $\frac{5x}{(x+2)(x-3)} + \frac{4}{(x+2)^2}$. Why don't we go any further?
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Continuity of solutions for systems of rational ODEs

If you have a system of ODEs of the from $\frac{d\mathbf{x}}{dt}=\mathbf{f(x,p)}$ where $\mathbf{x}$ is a vector valued variable, and $\mathbf{p}$ is a vector of parameters, and $\mathbf{f(x,p)}$ is a ...
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Showing that the field of rational functions is not dense

I am going through Counter Examples of Analysis but I am having trouble understanding a claim it makes. The book establishes that the set of rational functions defines an ordered field where the "...
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56 views

How would I divide $\frac{4x^2-2x}{x^2+5x+4}\div \frac{2x}{x^2+2x+1}$? [closed]

How would I divide $\frac{4x^2-2x}{x^2+5x+4}\div \frac{2x}{x^2+2x+1}$? Work would be appreciated. What would I do first?
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78 views

How do I multiply $\frac{8y-4}{10y-5}\times \frac{5y-15}{3y-9}$?

I am not sure how to multiply these fractions. Do I cross multiply or multiply the numerators together and multiply the denominators together? How do I multiply $\frac{8y-4}{10y-5}\times \frac{5y-15}{...
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1answer
237 views

The best way of integrating irrational functions

Ok, so, here is the example integral: $$I=\int\frac{x-2-\sqrt{-x^2-4x+4}}{x^2-\sqrt{-x^2-4x+4}}dx$$ I always solve these types of integrals using Euler's substitutions, but, recently, I came across ...
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1answer
28 views

Why is this a correct way to multiply 2 terms together?

In a problem, I noticed the author did this: $$\frac{1}{(a+2)+(z-2)} = \frac{1}{(a+2)}\cdot \frac{1}{1+\frac{z-2}{a+2}}$$ What he is saying is to take the entire $(a+2)$ term and multiply it by $1$ ...
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76 views

What is known about this group reminiscent of the anharmonic group?

The anharmonic group is this nonabelian group of six rational functions with the operation of composition of functions: \begin{align} t & \mapsto t & & \text{order 1} \\[8pt] t & \...
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181 views

Has this chaotic map been studied?

I have recently been playing around with the discrete map $$z_{n+1} = z_n - \frac{1}{z_n}$$ That is, repeatedly mapping each number to the difference between itself and its reciprocal. It shows some ...
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1answer
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$\{a x^{z}: a\in \Bbb{Q}, z \in \Bbb{Z}\} \approx \Bbb{Q}^{\times} \otimes_{\Bbb{Z}} \Bbb{Z}^+ \implies$? what about $\Bbb{Q}$-linear sums?

Consider all functions $f: \Bbb{Q} \to \Bbb{Q}$ of the form $f(x) = a x^z$ where $a \in \Bbb{Q}, z \in \Bbb{Z}$, call it $G$. It forms an abelian group under usual multiplication. I think it's ...