# Questions tagged [rational-functions]

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

1,263 questions
Filter by
Sorted by
Tagged with
23 views

34 views

### Horizontal asymptote of a Rational Function [duplicate]

I didn't quite understand how the horizontal asymptote of the function $r(x) = \frac{3x^2 - 2x - 1}{2x^2 + 3x - 2}$ can be $\frac{3}{2}$, since when you use desmos, for example, to graph it, part of ...
1k views

• 581
173 views

39 views

• 4,630
52 views

### Need to find the horizontal asymptotes of a given expression.

I need to determine the horizontal asymptotes of the following expression:$$\frac{2x^{1/3}}{(1x^2+4)^{1/6}}$$ I’m not even sure where to begin. I understand I need to divide all terms by the highest ...
• 33
99 views

### Simplifying rational function in Sage

Let $g\geq 1$ be an integer and define three rational functions and their sum by \begin{align*} H(q,t) &= f_1(q,t) + f_2(q,t) + f_3(q,t) \\ &:= \frac{t^{8g-4}q^{2g-1}(1+tq)^{2g-1}(1+q^2t^3)^{...
• 194
34 views

### Why can we discard remainders when calculating slant asymptotes of rational functions?

I understand how and when to calculate slant asymptotes of rational functions with numerators with one degree higher than the denominator, but I am confused as to why we can disregard the remainder ...
78 views

### At which points of $X=V(x^2+y^2-1)$ is the rational function $\frac{1-y}{x}$ regular?

Let $X \subseteq \mathbb{A}^2$ be the circle of equation $x^2+y^2=1$. I have to compute the points of $X$ such that $f=\frac{1-y}{x}$ is regular. (We can suppose $\operatorname{char}(K)\neq 2$) I ...
• 739
64 views

### Does Weierstrass substitution work for different arguments in the trig functions?

Suppose we have a two dimensional rational function $R(x,y)$. Then we can use Weierstrass substitution for following integral: $$\int R(\sin(x), \cos(x)) \, dx .$$ But can we use Weierstrass ...
• 389
1 vote
111 views

### Matching Requirements to a Rational Function

On a precalculus test I got recently, the following question stumped the entire class, and even weeks later, none of us could figure it out! The problem is to Write the equation of the following ...
• 11
31 views

### Splitting a black box rational function

Suppose I can evaluate a black box rational function $f(\vec{x})$. For the purposes of the explanation, let me take an example I can actually write down f(\vec{x})=\frac{x_1+x_2+x_3+...
• 375
79 views

### How to reliably find the global minimum of a rational function?

I am stuck at a algorithm problem. I was assigned the following problem: Given a finite Taylor expansion in some degree $n$: $$\text{TE} = \sum_{j=0}^{n}{c_j}{x^j}$$ with all real coefficients $c_j$ ...
• 31
54 views

I am reading "Introduction to Analysis" by Teiji Takagi. The author wrote the following proposition without a proof. Let $F(u,v)$ be a rational function. If $\pi$ is a period of $F(\cos x,\... • 1,106 0 votes 0 answers 30 views ### Partial derivative of rational function is identically zero Let$f:D\to\mathbb R$be the following rational function $$f(x_1,\ldots,x_n)=\frac{P(x_1,\ldots,x_n)}{Q(x_1,\ldots,x_n)},$$ where here$P,Q$are polynomials and $$D=\{(x_1,\ldots,x_n):Q(x_1,\ldots,... • 4,807 4 votes 0 answers 113 views ### Approximating \sqrt{x} by a rational function in the complex plane Newman (1963) proved the following. Theorem 1. Let d \in \mathbb{N}. Define$$p(x) = \prod_{k=0}^{d-1} \left(x+\exp\left(\frac{-k}{\sqrt{d}}\right)\right)$$and$$r(x) = \frac{\sqrt{x} \cdot (p(\... • 915 0 votes 1 answer 61 views ### Zorich Exercise 5.7.3: Reduction of integrals of the form$\int R(x,\sqrt{ax^2+bx+c})\,dx$Consider integrals of the form $$\int R(x,\sqrt{ax^2+bx+c})\,dx,\tag{1}$$ where$R$is a rational function of$x$and$\sqrt{ax^2+bx+c}$. Show that the integral$(1)$can always be reduced to ... • 137 2 votes 0 answers 39 views ### Is there a special name for symbolic ratios that share no common symbol in numerator and denominator I am wondering if there is a name for the operation$f$that separates a multivariate rational function into fractions, wherein neither numerator nor denominator share common symbols: given$y=\frac{x^...
• 465
Suppose that $f(x,z)$ and $g(x,y)$ are homogeneous rational functions of the 1st degree, that is, $f(tx,tz)=tf(x,z)$ and $g(tx,ty)=tf(x,y)$ and consider the function  F(x,y,z)=\big[f(x,z)+ay\big]\...