Consider the following sets of rational functions. af(x)=\frac{a}{x} for a={−2,−1,−0.5,0.5,2,4}

True or False. The graph of a rational function may intersect a horizontal asymptote.

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 ${\displaystyle V(x^2+y^2-1)\subseteq A^2}$ is the rational function ${\displaystyle \alpha =\frac{y-1}{x}}$ regular?

"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 "${\displaystyle \frac{p\left(x\right)}{q\left(x\right)}}$, where ${\displaystyle q\left(x\right)\ne 0}$."
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 ${\displaystyle x\mathrm{\&}{y}^{\prime }}$?

Suppose you have a rational function, of the form $H(x)=\frac{B(x)}{A(x)}$. It's differential is of the form $H^{\prime }(x)=\frac{(B^{\prime }(x)\star A(x)-B(x)\star A^{\prime }(x))}{(A(x){)}^2}$. It is trivial to prove that, if $k$ is a root of $B(x)$ or $A(x)$ with rank $r$, $r\ge 2$, then k is also a root of $H^{\prime }(x)$'s numerator, $B^{\prime }(x)\star A(x)-B(x)\star A^{\prime }(x)$, with rank at least $r-1$. Is there a theorem that states this exact thing?
The reason I need this is, I am going to take an exam soon, and in a specific part of the exam this observation would be very useful. But I will have to prove it by hand during the exam, and time will be very limited, so I was wondering if there is already a name for this to reference it directly and get on with it.
EDIT: Had erroneously named the desired form "polynomial fraction", instead of the right "rational function".

Do all rational functions have vertical asymptotes? Why or why not? If not, give an example of a rational function that does not have a vertical asymptote.

Find all the points of discontinuity of the function $\frac{y^2-18y+80}{(y-4)(y-1)}$.

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.
${\displaystyle \int \frac{1-\sqrt{x}}{1+\sqrt{x}}dx}$