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

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How to compute the limit of a rational function at infinity?

I am unable to compute the limit, anyone can help ? I only understand some basic knowledge of limit .
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Limit $\lim_{{x\rightarrow 0}}{\left(\frac{\frac{1}{\sqrt{1+x}}-1}{x}\right)}$

I am a high school student in Calculus, and we are finishing learning basic limits. I am reviewing for a big test tomorrow, and I could do all of the problems correctly except this one. I have no ...
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26 views

Rational function in both $k(X)[Y]$ and $k(Y)[X]$

If I have a rational function in $X$ and $Y$ and it can be written as both a polynomial in $Y$ with coefficients being rational functions in $X$ (that is, an element of $k(X)[Y]$) and as a polynomial ...
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1answer
40 views

Possible for variable to cancel in a product of multivariate rational expressions?

Let $f,g,p_i,q_i$ be polynomials over some field with $\gcd(p_i,q_i)=1$ and $q_i$ are not constants for $i=1,2$. Assume that one or more of $p_i$ or $q_i$ has a term containing a variable $x$ not ...
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32 views

Rational equality modulo $p$

Assume that we have two rational expressions $f,g\in \mathbb{Z}(x,y_1,\ldots,y_n)$ with the property that the variable $x$ can only appear in the numerator of $f$ while only in the denominator of $g$. ...
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41 views

Simplifying the sum of two rational functions

The original equation is: $$\frac{4x}{4x+1} + \frac{3}{16 x^2-1}$$ I am supposed to factorize then simplify it. Now after I factorized it I got $$\frac{4x(4x-1) + 3}{(4x+1)(4x-1)}$$ but I couldn't ...
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49 views

Find the domain of the function $f(x) =\frac{x+4}{x^2-9}$

I need to find the domain of the function $\;f(x) =\dfrac{x+4}{x^2-9}.$ My answer was: $(-\infty, -3)\cup(3, \infty)$. The book's answer was: $(-\infty, -3)\cup(-3,-3)\cup(3,+\infty)$ It's question ...
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Simplify rational expression

How do I simplfy this expression? $$\dfrac{\frac{x}{2}+\frac{y}{3}}{6x+4y}$$ I tried to use the following rule $\dfrac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}\cdot \frac{d}{c}$ But I did not get the ...
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3answers
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How to evaluate an integral of the form $\int \frac{dx}{-ax^2 + b}$?

I need to evaluate $\int \frac{dx}{-ax^2 + b}$ while both $a$ and $b$ are positive. I was blocked while I was trying $ x=\tan\theta $ which turned $ dx=\sec^2\theta\, d\theta $ This method didn't ...
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Revenue cost profit [closed]

A company that makes and sell memory chips establishes the followings : Revenue function, $R(x) = x (75 – 3x)$ Cost function, $C(x) = 125 + 16x$ Where $x$ is in millions of chips, and $R(x)$ and ...
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249 views

Limit of a rational function

Calculate the limit $$ \lim_{x \to 0} \frac{3x^{2} - \frac{x^{4}}{6}}{(4x^{2} - 8x^{3} + \frac{64x^{4}}{3} )}$$ I divided by the highest degree of x, which is $x^{4}$, further it gave $$ ...
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50 views

Integral of $\arcsin$ of a rational function, using integration by parts

I'm a class 12 student and this a question from my textbook: $$I=\int{\arcsin{2x\over 1+x^2}}\mathrm{d}x$$ I did it using integration by parts like this: $$I=\arcsin{\left(2x\over ...
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52 views

A simple-looking rational limit

Please help me compute: $$ \lim_{z\to 0}\frac{\sqrt{2(z-\log(1+z))}}{z} $$ I know the answer is 1 because I plugged it into Mathematica. Attempts with L'Hopital's Rule didn't work. This a step in an ...
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70 views

Find all values $c$ such that $(x+1)/(x^2+2cx+4)$ has domain R

Word for word: Find all values of $c$ such that $f(x)=\frac{x+1}{x^2+2cx+4}$ has a domain R I really don't know where exactly to start. I'm not sure what it means by "a domain R"
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38 views

Why is this rational expressions indeterminate when evaluated?

I have this rational expression to evaluate, $$ {{3a-3}\over {4a(a-1)}} \text { if } a=1. $$ I understand that if you substitute 1, both the numerator and denominator would turn out 0, thus making ...
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51 views

Integration of the rational function $ 4/( 1+4t^2) $

So it's August so my memory of math is a little rough right now. I was wondering if someone could help me with integration with a fraction involved? For example: $$\int_0^{1/2} \frac 4{1+4t^2} \, ...
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1answer
34 views

Is there any rational map from the nonsingular Segre quadric surface in $\mathbb{CP}^3$ to a nonsingular surface of degree greater or equal to 4?

Is there any rational map from the nonsingular Segre quadric surface in $\mathbb{CP}^3$ to a nonsingular surface in $\mathbb{CP}^3$ of degree greater or equal to 4? Someone told me that the answer is ...
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5answers
150 views

Why is $f(n) =\frac{n(n+1)(n+2)}{(n+3)}$ in $O(n^2)$?

Let: $$f(n) = n(n+1)(n+2)/(n+3)$$ Therefore : $$f∈O(n^2)$$ However, I don't understand how it could be $n^2$, shouldn't it be $n^3$? If I expand the top we get $$n^3 + 3n^2 + 2n$$ and the biggest ...
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Neither $\log x$ nor $\exp(x)$ are rational functions [closed]

(a) Prove that $\log x$ cannot be expressed in the form $f(x)/g(x)$ where $f(x)$ and $g(x)$ are polynomials with real coefficients. (b) Prove that $e^x$ cannot be expressed in the form $f(x)/g(x)$ ...
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51 views

Integral of an exponential of rational function

I have an integral of the form $\int_{a}^{b} \text{exp}\left(\frac{\lambda}{\rho^2 m + \sigma^2_u}\right) \frac{1}{m^2}\text{exp}\left(-\frac{\lambda}{m}\right) dm$. Can this integral be found ...
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1answer
79 views

Why does the graph of $x^3/x^3$ not have a horizontal asymptote?

I am a graduate student studying math, and am actually teaching College Algebra right now. But every once in a while, I come upon something new in a subject that I have supposedly mastered. Why does ...
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36 views

Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials): (a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$; (b) the first $k$ ...
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31 views

Variation of the argument of a rational function along a circle

Crossposted here on MO. Let $f:\bar{\mathbb C}\to \bar{\mathbb C}$ be a rational function, and take a circle $C$ not crossing the zero- and polar-locus of $f$. The argument principle tells us the ...
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40 views

Help solve rational expression

I need help solving this rational expression. Divide $$\frac{4x^4 + 6x^3 + 3x - 1}{2x^2 + 1}$$ How do you solve this problem? Where do I start?
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Setting up word problem for finding length and width

Word Problem: The length of a rectangular sign is $3$ feet longer than the width. If the sign has space for $54$ square feet of advertising, find its length and width. I have not idea where to start. ...
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44 views

Rationals over an interval

Suppose $I$ is an interval $[a,b]$. It is noted that $a$ and $b$ are real integers. Divide the interval into $n$ parts with step size $h=(b-a)/n$. Clearly all the points $a$, $a+h$, ...
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Solving the polynominal: $s(t) = -16t^2 + 48t + 160$

The height of a ball is thrown directly upward from an initial height of $160$ ft with an initial velocity of $48$ ft per second is given by the function: $s(t) = -16t^2 + 48t + 160$, where $s(t)$ ...
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Solving for $x$: $\;\frac8{x-2}-\frac{13}2=\frac3{2x-4}.$

I have solved this sort of problem before, but frustratingly I have forgotten to. The problem is: Solve for $x$: $$\dfrac8{x-2}-\dfrac{13}2=\dfrac3{2x-4}.$$ So hence the title, How should I ...
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The value of $a/b+b/a-ab$ under the condition $ab=a-b$ (AMC 10 problem)

I am confused about the problem below. I answered correctly but used substitution to solve instead of changing everything to the same denominator as shown in the solution. The problem: Two non-zero ...
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Derivative of rational function help.

consider $$f(x)=\frac{1}{2x-4}$$ The derivative should be $\displaystyle -\frac{1}{2(2x-4)^2}$ However I get $\displaystyle -\frac{2}{(2x-4)^2}$ my workflow: $$\begin{array}{} f'(x)&= ...
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1answer
40 views

Supremum of a rational function

Let $f(z)$ be a rational function in the complex plane such that $f$ does not have any poles in $\{z:\Im z\ge0\}$. Prove that $\sup\{|f(z)|:\Im z\ge0\}=\sup\{|f(z)|:\Im z=0\}$. Let $\Gamma_r$ be a ...
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There cannot exist a rational function $f: \mathbb{R} \to \mathbb{R}$ injective, not surjective

I was looking for a rational function $f: \mathbb{R} \to \mathbb{R}$ that looks like $\arctan$, in that it is injective not surjective well-defined on all $x\in \mathbb{R}$ (no vertical ...
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Where rational functions are undefined

I have another question/comment I'd like a fresh pair of eyes on The question is "A rational function can have infinitely many x-values at which it is not continuous" I know since Q(x) in the ...
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min max of a rational function

The variable is a vector $x \in \mathbb{R}^n \times\mathbb{R}^m_+ \times \mathcal{E}$ where $\mathcal{E}$ is an ellipsoid of dimension $e$. I would like to find the min and max of the following ...
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Rational function on a noetherian scheme

Let $X$ be a noetherian scheme and $f$ a rational function on $X$, so by definition the domain of $X$ includes all associated points of $X$. I think the following is true: $f$ is regular on $X$ if and ...
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The solutions for the equation $\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$

How can I find the solution for the following equation in $a,b \mbox{ and } c$. $$\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$$ Also $b-c \neq 1$, $c-a \neq 1$ and $a-b \neq 1$. Thanks!
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Branch of mathematics that studies groups / rings or rational functions

I'm not really a mathematician, and looking for some literature which could potentially help me in research. Im interested in algebra of rational functions (ratios of polynomials) of finite order. ...
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1answer
19 views

Termwise differentiation of sequence of rational functions when the uniform limit is analytic

Given a sequence $\{f_n(x)\}$ of rational functions which converges uniformly to the analytic function $\{g(x)\}$ on $[a, b]$ ($f_n(x)$ are defined on $[a, b]$ and hence are analytic), what can we say ...
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Inverse functions problem

There are two functions $f\colon\mathbb Q \to \mathbb Q \setminus \{-1\}$ and $g\colon\mathbb Q \to \mathbb Q \setminus \{1\}$. $$g(x) = \frac{f(x)}{f(x)+1}.$$ Prove that if there is a inverse ...
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Values of $m$ for which $y^2 + 2xy + 2x -my -3$ can be factorised

For what values of $m$, will the expression $y^2 + 2xy + 2x -my -3$ be capable of resolution into two rational linear factor? This is how I did it: $$y^2 + 2xy + 2x -my -3 = y^2+(2x-m)y+2x-3$$ ...
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1answer
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rational parameterization of quartic

With the curve $x^4 - 6x^2 - y^2 + 1 = 0$ in the range of $x$ inside of $(-1,1)$, I can only identify two rational points $(0,1)$ and $(0,-1)$. Is it possible to determine if there are others?
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Inverse LaPlace Transform of the square root of Rational, Monic 1st Degree Polynomials

I tried to find this in Churchill's Operational Mathematics which has a good variety of transform pairs, but no matches for what appears a simple expression. Does anyone have a solution for the ...
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1answer
42 views

Rational parametrization of points on sphere of irrational radius

I am trying to figure out if a closed form parametrization can exist for finding all of the rational points on a sphere of radius $\sqrt{2}$. ie, x=f(u,v), y=g(u,v), and z=h(u,v) where f,g,and h are ...
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integrality of certain rational numbers

Let $P,Q\in\mathbb Q[X]$ be relatively prime polynomials ($X$ being an indeterminate). Assume that $Q(0)=1$ and that $P/Q$ is in $\mathbb Z[[X]]$. Does this imply that $P$ and $Q$ are in $\mathbb ...
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18 views

recursive relation in rational expression form

I am looking for a closed form expression for the variables $n_i$ that are stationary solutions of the recursive relation: $ n_i(t+1)=n_i(t)\sum_mf_{i,m}\frac{K_m}{\sum_j f_{j,m}n_j(t)}$ i.e. the ...
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81 views

expand a rational function in a power series

$$\frac{4-x}{(2-x)(1-x)^2}$$ Expand in ascending powers of x, stating when the expansion is valid; also write down the coefficient of $x^n $
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66 views

Solving $L= \frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ priveded $a+b+c=0$

Let $a,b,c$ be such that $a+b+c=0$ and suppose that $$L= \frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}.$$ Find the value of $L$. I can only see the symmetry of these function ...
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1answer
78 views

Find the sum of the maximum and minimum

For a real number $x$ find the sum of the maximum and minimum. $$y=\frac{x^2-2x-3}{2x^2+2x+1}$$ This is a sample question for a math competition. It seems like calculus would be used to solve this, ...
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83 views

Geometry aspect of a extreme value problem

In a plain with orthogonal coordinate $XOY$, set point $A(a,a)$, and $P$ is a point in function $y=\frac{1}{x}$,where $x>0$. If the distance between $P$ and $A$ is $2\sqrt{2}$.Find all $a$ ...
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1answer
255 views

How to prove $\frac{1}{x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+2\sqrt{\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}}$

Question: Let $a,b,c>0$ are give numbers and $x>0$, such that $$ \sqrt{\dfrac{a+b+c}{x}}=\sqrt{\dfrac{b+c+x}{a}}+\sqrt{\dfrac{c+a+x}{b}}+\sqrt{\dfrac{a+b+x}{c}} $$ show that $$ ...