The line $y = 5$ is a horizontal asymptote to the graph of which of the following functions?
(A) $y = \frac { \sin ( 5 x ) } { x }$
(B) $y = 5 x$
(C) $y = \frac { 1 } { x - 5 }$
(D) $y = \frac { 5 x } { 1 - x }$
(E) $y = \frac { 20 x ^ { 2 } - x } { 1 + 4 x ^ { 2 } }$

The line $y = 5$ is a horizontal asymptote to the graph of which of the following functions?
(A) $y = \frac { \sin ( 5 x ) } { x }$
(B) $y = 5 x$
(C) $y = \frac { 1 } { x - 5 }$
(D) $y = \frac { 5 x } { 1 - x }$
(E) $y = \frac { 20 x ^ { 2 } - x } { 1 + 4 x ^ { 2 } }$

In the plane equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, we denote by $\mathscr{C}_{u}$ the curve representing the function $u$ defined on the interval $]0; +\infty[$ by: $$u(x) = a + \frac{b}{x} + \frac{c}{x^2}$$ where $a, b$ and $c$ are fixed real numbers.
The curve $\mathscr{C}_{u}$ passes through the points $\mathrm{A}(1; 0)$ and $\mathrm{B}(4; 0)$ and the $y$-axis and the line $\mathscr{D}$ with equation $y = 1$ are asymptotes to the curve $\mathscr{C}_{u}$.

1. Give the values of $u(1)$ and $u(4)$.
2. Give $\lim_{x \rightarrow +\infty} u(x)$. Deduce the value of $a$.
3. Deduce that, for all strictly positive real $x$, $u(x) = \frac{x^2 - 5x + 4}{x^2}$.

The English physiologist Archibald Vivian Hill proposed, in his studies, that the velocity $v$ of contraction of a muscle when subjected to a weight $p$ is given by the equation $( p + a )( v + b ) = K$, with $a$, $b$, and $K$ constants.
A physiotherapist, with the intention of maximizing the beneficial effect of the exercises he would recommend to one of his patients, wanted to study this equation and classified it as follows:

Type of curve

Oblique half-line

Horizontal half-line

Branch of parabola

Arc of circle

Branch of hyperbola

The physiotherapist analyzed the dependence between $v$ and $p$ in Hill's equation and classified it according to its geometric representation in the Cartesian plane, using the coordinate pair ($p$; $v$). Assume that $K > 0$.
The graph of the equation that the physiotherapist used to maximize the effect of the exercises is of the type
(A) oblique half-line.
(B) horizontal half-line.
(C) branch of parabola.
(D) arc of circle.
(E) branch of hyperbola.

Let $$f ( x ) = \frac { x ^ { 4 } } { ( x - 1 ) ( x - 2 ) \cdots ( x - n ) }$$ where the denominator is a product of $n$ factors, $n$ being a positive integer. It is also given that the $X$-axis is a horizontal asymptote for the graph of $f$. Answer the independent questions below by choosing the correct option from the given ones. a) How many vertical asymptotes does the graph of $f$ have?
Options: $n$ less than $n$ more than $n$ impossible to decide
Answer: $\_\_\_\_$ b) What can you deduce about the value of $n$?
Options: $n < 4$ $n = 4$ $n > 4$ impossible to decide
Answer: $\_\_\_\_$ c) As one travels along the graph of $f$ from left to right, at which of the following points is the sign of $f ( x )$ guaranteed to change from positive to negative?
Options: $x = 0$ $x = 1$ $x = n - 1$ $x = n$
Answer: $\_\_\_\_$ d) How many inflection points does the graph of $f$ have in the region $x < 0$?
Options: none 1 more than 1 impossible to decide
(Hint: Sketching is better than calculating.)
Answer: $\_\_\_\_$

Write your answers to each question below as a series of three letters Y (for Yes) or N (for No). Leave space between the group of three letters answering (i), the answers to (ii) and the answers to (iii). Consider the graphs of functions $$f(x) = \frac{x^{3}}{x^{2}-x} \qquad g(x) = \frac{x^{2}-x}{x^{3}} \qquad h(x) = \frac{x^{3}-x}{x^{3}+x}$$ (i) Does $f$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?
(ii) Does $g$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?
(iii) Does $h$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?

$$f ( x ) = \frac { - k x ^ { 3 } + k ^ { 2 } x } { k ^ { 3 } x ^ { 2 } + x - ( k + 1 ) }$$
The function has a vertical asymptote at $x = 1$. Accordingly, what is the value of $f ( 2 )$?
A) - 5
B) - 4
C) - 3
D) - 2
E) - 1

The function
$$f ( x ) = \frac { a x } { | b x + 2 | }$$
defined on a subset of the set of positive real numbers has a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = 4$.
Accordingly, what is the sum $a + b$?
A) 1
B) 2
C) 3
D) 4
E) 5

Let a be a real number, and $$f ( x ) = \ln ( 2 x + 8 )$$ The vertical asymptote of the function $$g ( x ) = \frac { \sin x } { x ^ { 2 } + a x }$$ is also a vertical asymptote of the function.\ Accordingly, what is a?\ A) 0\ B) 1\ C) 2\ D) 3\ E) 4