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$lim_{x \to 2} \frac{\sqrt{x + 2} \sqrt{x + 7} - 2 \sqrt{x + 7} - 3 \sqrt{x + 2} + 6}{x \sqrt{4 x + 1} - 2 \sqrt{4 x + 1} - 3 x + 6}$
How to solve this???

Guinzio3rv 2022-02-17

All polynomials are rational functions, that perhaps I should just imagine them as "over 1". Good. But the definition of a rational function has the concept of "ratio" in it. And when I found this:
A cylinder has a volume of $(x + 3) (x^{2} - 3 x - 18) π$ cubic centimeters. Find the height of the cylinder.
I wondered how this is a ratio of any sort?(It can't be a ratio of the expression over 1) So if $a = (x + 3), b = (x^{2} - 3 x - 18)$ , and $c = π$ ,
can we say a,b,c are "in a ratio," i.e., a:b:c?

Romana Newton 2022-02-16

Does it follow that $r (x) + p (x)$ is a rational function (and not a polynomial)?
Suppose $r (x) \neq 0$ is a rational function (and not a polynomial)
$p (x)$ is a polynomial function of $x \in R$ .

Naeem Stanton 2022-02-16

I know that all meromorphic functions on $C P P^{1}$ are rational functions. However, Im

Guinzio3rv 2022-02-16

Suppose $R (z)$ be a rational function such that |R(z)|=1 for |z|=1. Show that $α$ is a zero or a pole or order $m$ , if and only if $\frac{1}{\overset{―}{α}}$ is a pole or zero or order $m P$ respectively.
I am wondering that if I can first show that $M (z) = R (z)$ the complex conjugate of $R (\frac{1}{z})$ is a rational function such that $M (z) = 1 P$ on $| z | = 1$ . Can I assume $z = 1$ and then substitute all $z$ with 1? Thanks.

kungezwi8af 2022-02-16

I thought it was interesting that $\frac{u^{2} + 1}{{(u^{2} - 2 u - 1)}^{2}}$ has the very simple integral $- \frac{u}{u^{2} - 2 u - 1}$ but noth of ${\frac{u^{2}}{(u^{2} - w u - 1)}}^{2}$ and $\frac{1}{{(u^{2} - 2 u - 1)}^{2}}$ are very complicated (the transcendental parts cancel each other though).
So my question is how do I check by looking at a rational function whether or not its

Akash Matthews 2022-02-16

"determine the form of rational function in a plane hat has a positive value on unit circle."
hint suggested me that such a rational function must have same number of zeros and poles inside unit disk. But I cannot understand that. Using symmetry(Schwarz reflection principle) I can see that number of poles(zeros) inside unit disk is equal to number of poles(zeros) outside unit disk. BUT I CANNOT SEE WHY NUMBER OF POLES AND ZEROS MUST BE EQUAL..
Who knows?

Bruce Partridge 2022-02-16

Help me with creating a rational function with vertical asymptotes $x = \pm 1$ and oblique asymptote of $y = 2 x - 3$ and a $y$ -intercept of 4.

Guy Rollins 2022-02-16

I have to find border of rational function:
$lim_{x \to \infty} \frac{x^{3} + 3 x^{2} + 2 x}{x^{2} - x - 6}$
What should I do?

Cormac Gilbert 2022-02-16

For the function, determine if it is a polynomial, rational function, both, or neither
$g (x) = \frac{4 x^{5} + x^{2}}{- 8 x^{9} + 2 \sqrt{x} + 9}$
a. Polynomial function only
b. Rational function only
c. Both polynomial and rational function

Miruna Atherton 2022-02-16

Construct all rational functions f (quotients of polynomials) in $C$ such that $(f) = 6 [0] - 4 [5]$ , where (f) is the divisor of the function f.
My answer/attempt: From my understanding of divisors, there seems to be exactly one function, namely, $f (z) = \frac{z^{6}}{{(z - 5)}^{4}}$ .
1.Are there any other functions with this divisor? If not, why?
2.If we restrict ourselves to divisors of rational functions only, then if we are given a divisor of a $△$ on $C$ , does there exists a unique rational function f with $(f) = △$ ?

enfocarteu7z 2022-02-16

Lagrange's rational function theorem states that if one has two rational functions in multiple variables $f (x_{1}, x_{2}, \dots x_{n})$ and $g (x_{1}, x_{2}, \dots x_{n})$ then one can express f as a rational function in g if and only if the set of permutations that keep g unchanged is a subset of the set of permutations that preserve f.
Is anyone familiar with the proof of this theorem? While it is fairly clear that if f can be expressed in terms of g the set of permutations that keep g unchanged has to be the subset of those that keep f unchanged, the converse is far from obvious.

sunyerneq 2022-02-16

In Hartshorne's Algebraic Geometry the function field K(Y) of a variety Y is defined as the set of equivalence classes with f being a regular function on the open subset $U \neq \emptyset$ . We have = if f and g agree on $U \cap V$ .
How does one evaluate a "rational function on Y'' f? Is it f(P) using the Evaluation map for polynomials? In this case, I don't see that this is well defined: Let $\frac{P \in U}{V}$ and =, then g(P) might not even be defined!

Havlishkq 2022-02-16

What is rational function and how do you find domain, vertical and horizontal asymptotes. Also what is "holes" with all limits and continuity and discontinuity?

enfocarteu7z 2022-02-16

A polynomial with integer coefficients maps integers to integers. A rational function (which is a quotient of two polynomials) tends to map integers to rational numbers. Other than the trivial case where the denominator is a factor of the numerator, are there any rational functions that map all integers to integers? What is known about them and how can they be found?

Bradlee Hooper 2022-02-15

"A rational function is defined as the quotient of polynomials in which the denominator has a degree of at least 1"
If we are talking merely about x, then I get the concept. A rational function f(x) could be written as " $\frac{p (x)}{q (x)}$ , where $q (x) \neq 0$ ."
The issue that I'm having is that of talking about rational functions of n variables. For instance, what would be the meaning of 'f(x,y) is a rational function of $x & y^{'}$ ?

SnuluddidaPalbx4 2022-02-15

Let $f : X \to P^{1}$ be a rational function of degree $d \geq 2$ on a curve X.
Let $n \geq 2$ be a divisor of d. Does there exist a curve Y with a rational function $g : Y \to P^{1}$ of degree n such that f factors through g
Can we factor f in some sense?
Note that by de Franchis

Brittney Cuevas 2022-02-15

a) Show sin is not a rational function.
By definition of a rational function, a rational function cannot be 0 at infinite points unless it is 0 everywhere. Obviously, sin have infinite points that are 0 and infinite points that are not zero, thus not a rational function.
b) Show that there do not exist rational functions $f_{0}, \dots, f^{n - 1} (x) {(\sin x)}^{n - 1} + \dots + f_{0} (x) = 0$ for all x
First choose $x = 2 k π$ , so $f_{0} (x) = 0$ for $x = 2 k π$ . Since $f_{0}$ is rational $\Rightarrow f_{0} (x) = 0$ for all x. Thus can write $\sin x [{(\sin x)}^{n - 1} + f_{n - 1} {(\sin x)}^{n - 2} \dots + f_{1} (x)] = 0$
Question: The second factor is 0 for all $x \neq 2 k π$ How does this imply that it is 0 for all x? And how does this lead to the result?

Casertowm7 2022-02-15

In general, how does one determine if a rational function is regular? I have the particular problem of determining in which points of the circle $V (x^{2} + y^{2} - 1) \subseteq A^{2}$ is the rational function $α = \frac{y - 1}{x}$ regular?

Ausanioyck 2022-02-15

For each of the following functions, determine if it is a polynomial, rational function, both, or neither
$h (x) = 3 x^{- 9} + 6 x^{- 4} - 3$
a. Polynomial function only
b. Rational function only
c. Both polynomial and rational function

$j (x) = {(8 x^{2} - 5 \sqrt{x} + 6)}^{2}$
a. Polynomial function only
b. Rational function only
c. Both polynomial and rational function

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