$\lim _ { x \rightarrow 0 } \frac { 1 - \cos ^ { 2 } ( 2 x ) } { ( 2 x ) ^ { 2 } } =$
(A) 0
(B) $\frac { 1 } { 4 }$
(C) $\frac { 1 } { 2 }$
(D) 1

O limite $\displaystyle\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}$ é igual a
(A) 0 (B) 1 (C) 2 (D) 4 (E) indefinido

The value of $\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } - 1 } { \sqrt { x + 3 } - 2 }$ is? [2 points]
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11

Find the value of $\lim _ { x \rightarrow 1 } \frac { ( x - 1 ) \left( x ^ { 2 } + 3 x + 7 \right) } { x - 1 }$. [3 points]

What is the value of $\lim _ { x \rightarrow 0 } \frac { \ln ( 1 + 5 x ) } { \sin 3 x }$? [2 points]
(1) 1
(2) $\frac { 4 } { 3 }$
(3) $\frac { 5 } { 3 }$
(4) 2
(5) $\frac { 7 } { 3 }$

For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Explain why, for every $x > 0$, there exists a $y_1 \in \left]0, 1\right[$ such that $$(\delta(f))(x) = f'(x + y_1)$$

111- If $2^a = \displaystyle\lim_{x \to \frac{\pi}{4}} \dfrac{\sqrt{\cos x} - \sqrt{\sin x}}{\cos(x + \frac{\pi}{4})}$, then what is $a$?
(1) $-\dfrac{1}{2}$ (2) $-\dfrac{1}{4}$ (3) $\dfrac{1}{4}$ (4) $\dfrac{1}{2}$

114- What is $\displaystyle\lim_{x \to \pi} \dfrac{\sin(1 + \cos x)}{1 - \cos 2x}$?
(1) $\dfrac{1}{4}$ (2) $\dfrac{1}{2}$ (3) $1$ (4) $2$

111- What is $\displaystyle\lim_{x\to 0}\dfrac{\cos^2 x - \sqrt{\cos x}}{x^2}$?
(1) $-\dfrac{3}{2}$ (2) $-\dfrac{3}{4}$ (3) $-\dfrac{1}{4}$ (4) $\dfrac{3}{2}$
%% Page 20 Mathematics 120-C Page 3

111- The limit of $\dfrac{\sqrt{\cos 3x} - \sqrt{\cos x}}{x^2}$ as $x \to 0$ is which of the following?
(1) $-2$ (2) $-\dfrac{1}{2}$ (3) $\dfrac{1}{2}$ (4) $2$

The limit $\lim \left[ \left\{ 1 - \cos \left( \sin ^ { 2 } a x \right) \right\} / x \right]$ as $x -> 0$
(a) Equals 1
(b) Equals a
(c) Equals 0
(d) Does not exist

20. $\lim x \rightarrow 0 ( x \tan 2 x - 2 x \tan x ) / ( 1 - \cos 2 x ) 2$ is:
(A) $y = 2$
(B) $y = 2 x$
(C) $y = 2 x - 4$
(D) $y = 2 \times 2 - 4$

22. The integer $n$ for which $\lim _ { x \rightarrow 0 } ( \cos x - 1 ) \left( \cos x - e ^ { x } \right) / x ^ { n }$ is a finite non-zero number is
(A) 1

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(B) 2
(C) 3
(D) 4

9. If $\lim _ { ( x \rightarrow 0 ) } ( ( ( a - n ) n x - \tan x ) \sin n x ) / x ^ { 2 } = 0$, where $n$ is non zero real number, then $a$ is equal to:
(a) 0
(b) $( n + 1 ) / n$
(c) $n$
(d) $n + 1 / n$

The largest value of the nonnegative integer $a$ for which $$\lim_{x \rightarrow 1} \left\{\frac{-ax + \sin(x-1) + a}{x + \sin(x-1) - 1}\right\}^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4}$$ is

Let $m$ and $n$ be two positive integers greater than 1 . If
$$\lim _ { \alpha \rightarrow 0 } \left( \frac { e ^ { \cos \left( \alpha ^ { n } \right) } - e } { \alpha ^ { m } } \right) = - \left( \frac { e } { 2 } \right)$$
then the value of $\frac { m } { n }$ is

Let $\alpha, \beta \in \mathbb{R}$ be such that $\lim_{x \rightarrow 0} \frac{x^2 \sin(\beta x)}{\alpha x - \sin x} = 1$. Then $6(\alpha + \beta)$ equals

Let $k \in \mathbb { R }$. If $\lim _ { x \rightarrow 0 + } ( \sin ( \sin k x ) + \cos x + x ) ^ { \frac { 2 } { x } } = e ^ { 6 }$, then the value of $k$ is
(A) 1
(B) 2
(C) 3
(D) 4

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

The value of $\lim_{x \rightarrow 0} \frac{(1 - \cos 2x)(3 + \cos x)}{x \tan 4x}$ is equal to
(1) 1
(2) 2
(3) $-\frac{1}{4}$
(4) $\frac{1}{2}$

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

$\lim _ { x \rightarrow 0 } \frac { ( 1 - \cos 2 x ) ( 3 + \cos x ) } { x \tan 4 x } =$
(1) $\frac { 1 } { 2 }$
(2) 4
(3) 3
(4) 2

$\lim_{x \to \pi/2} \frac{\cot x - \cos x}{(\pi - 2x)^3}$ equals:
(1) $\frac{1}{24}$
(2) $\frac{1}{16}$
(3) $\frac{1}{8}$
(4) $\frac{1}{4}$

$\lim_{x \to \pi/2} \frac{\cot x - \cos x}{(\pi - 2x)^3}$ equals: (1) $\frac{1}{24}$ (2) $\frac{1}{16}$ (3) $\frac{1}{8}$ (4) $\frac{1}{4}$

$\lim _ { x \rightarrow 0 } \frac { ( 1 - \cos 2 x ) ^ { 2 } } { 2 x \tan x - x \tan 2 x }$ is
(1) 2
(2) $- \frac { 1 } { 2 }$
(3) $- 2$
(4) $\frac { 1 } { 2 }$