Let $f ( x ) = \left| \frac { \sin x } { x } \right| ^ { 1.001 }$ for $x \neq 0$ and $f ( 0 ) = L$ such that $f$ is continuous. Let $I ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
Statements
(21) $L = 1.001$ (22) $I ( 0.001 ) > 0.001$. (23) As $x \rightarrow \infty$ the limit of $I ( x )$ is greater than 1001 (possibly $\infty$). (24) The function $I ( x )$ is NOT differentiable at infinitely many points.

What is the value of $\lim_{x \rightarrow 0} \frac{3x^{2}}{\sin^{2} x}$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Show that $\lambda ^ { \alpha } \leq 1 + \lambda$ for all real $\lambda > 0$ and deduce the inequality:
$$\left| x - \frac { k } { n } \right| ^ { \alpha } \leq n ^ { - \alpha / 2 } \left( 1 + \sqrt { n } \left| x - \frac { k } { n } \right| \right)$$
for all $x \in ]0,1[ , n \in \mathbb { N } ^ { * }$ and $k \in \{ 0 , \ldots , n \}$.

Show that there exists a real $a > 0$ such that $$\forall \theta \in [-\pi,\pi], 1-\cos\theta \geq a\theta^2.$$ Deduce that there exist three reals $t_0 > 0$, $\beta > 0$ and $\gamma > 0$ such that, for all $t \in ]0,t_0]$ and all $\theta \in [-\pi,\pi]$, $$\left|\frac{P(e^{-t}e^{i\theta})}{P(e^{-t})}\right| \leq e^{-\beta(t^{-3/2}\theta)^2} \quad \text{or} \quad \left|\frac{P(e^{-t}e^{i\theta})}{P(e^{-t})}\right| \leq e^{-\gamma(t^{-3/2}|\theta|)^{2/3}}.$$

By taking $t = \frac{\pi}{\sqrt{6n}}$ in formula $$p_n = \frac{e^{nt} P(e^{-t})}{2\pi} \int_{-\pi}^{\pi} e^{-in\theta} \frac{P(e^{-t}e^{i\theta})}{P(e^{-t})} \mathrm{d}\theta,$$ conclude that $$p_n = O\left(\frac{\exp\left(\pi\sqrt{\frac{2n}{3}}\right)}{n}\right) \quad \text{when } n \text{ tends to } +\infty.$$

$\lim _ { x \rightarrow 0 } \frac { \left( e ^ { x } - 1 \right) \tan ^ { 2 } x } { x ^ { 3 } }$
(a) does not exist
(b) exists and equals 0
(c) exists and equals $\frac { 2 } { 3 }$
(d) exists and equals 1.

Let $f ( x ) = \frac { 1 } { 2 } x \sin x - ( 1 - \cos x )$. The smallest positive integer $k$ such that $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { k } } \neq 0$ is:
(A) 3
(B) 4
(C) 5
(D) 6.

6. Determine the real number $a$ so that the value of
$$\lim _ { x \rightarrow 0 } \frac { \sin ( x ) - x } { x ^ { a } }$$
is a non-zero real number.

Determine the values of the real parameters $a$ and $b$ so that:
$$\lim_{x \rightarrow 0} \frac{\operatorname{sen} x - (ax^{3} + bx)}{x^{3}} = 1$$

$\lim _ { x \rightarrow 0 } \frac { \sin \left( \pi \cos ^ { 2 } x \right) } { x ^ { 2 } }$ is equal to
(1) $- \pi$
(2) $\pi$
(3) $\frac { \pi } { 2 }$
(4) 1

A vector $\vec { A }$ is rotated by a small angle $\Delta \theta$ radians ( $\Delta \theta \ll 1$ ) to get a new vector $\vec { B }$. In that case $| \vec { B } - \vec { A } |$ is :
(1) $| \vec { A } | \left[ 1 - \frac { ( \Delta \theta ) ^ { 2 } } { 2 } \right]$
(2) 0
(3) $| \vec { A } | \Delta \theta$
(4) $| \vec { B } | \Delta \theta - | \vec { A } |$

$$\lim _ { x \rightarrow 0 } \frac { \sin 3 x } { 2 - \sqrt { 4 - x } }$$
What is the value of this limit?
A) 3
B) 9
C) 12
D) 15
E) 16

$$\lim _ { x \rightarrow 1 ^ { + } } ( x - 1 ) \cdot \ln \left( x ^ { 2 } - 1 \right)$$
What is the value of this limit?
A) $\frac { -1 } { 2 }$
B) $-2$
C) 0
D) 1
E) 4

For a function f defined on the set of real numbers
$$\begin{aligned} & \lim _ { x \rightarrow 3 ^ { + } } f ( x ) = 1 \\ & \lim _ { x \rightarrow 3 ^ { - } } f ( x ) = 2 \end{aligned}$$
Given this, what is the value of the limit $\lim _ { x \rightarrow 2 ^ { + } } \frac { f ( 2 x - 1 ) + f ( 5 - x ) } { f \left( x ^ { 2 } - 1 \right) }$?
A) $\frac { -1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) 1
D) 3
E) 4