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

$$f ( x ) = \frac { x } { x + \sin x } \quad \text { and } \quad g ( x ) = \frac { x ^ { 4 } + x ^ { 6 } } { e ^ { x } - 1 - x ^ { 2 } }$$
(a) Limit as $x \rightarrow 0$ of $f ( x )$ is $\frac { 1 } { 2 }$.
(b) Limit as $x \rightarrow \infty$ of $f ( x )$ does not exist.
(c) Limit as $x \rightarrow \infty$ of $g ( x )$ is finite.
(d) Limit as $x \rightarrow 0$ of $g ( x )$ is 720.

$\lim _ { x \rightarrow a } \frac { 2 ^ { x } - 1 } { 3 \sin ( x - a ) } = b \ln 2$ is satisfied by two constants $a , b$. What is the value of $a + b$? [3 points]
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 1 } { 2 }$

(Calculus) Point $\mathrm{A}$ is on circle $\mathrm{O}$ with radius 1. As shown in the figure, for a positive angle $\theta$, two points $\mathrm{B}$ and $\mathrm{C}$ on circle $\mathrm{O}$ are chosen such that $\angle \mathrm{BAC} = \theta$ and $\overline{\mathrm{AB}} = \overline{\mathrm{AC}}$. Let $r(\theta)$ denote the radius of the inscribed circle of triangle $\mathrm{ABC}$. If $\lim_{\theta \rightarrow \pi - 0} \frac{r(\theta)}{(\pi - \theta)^2} = \frac{q}{p}$, find the value of $p^2 + q^2$. (Given: $p$ and $q$ are coprime natural numbers.) [4 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) $\frac { 7 } { 3 }$

As shown in the figure, there is a rhombus ABCD with side length 1. Let E be the foot of the perpendicular from point C to the extension of segment AB, let F be the foot of the perpendicular from point E to segment AC, and let G be the intersection of segment EF and segment BC. If $\angle \mathrm { DAB } = \theta$, let the area of triangle CFG be $S ( \theta )$.
What is the value of $\lim _ { \theta \rightarrow 0 + } \frac { S ( \theta ) } { \theta ^ { 5 } }$? (Here, $0 < \theta < \frac { \pi } { 2 }$) [4 points]
(1) $\frac { 1 } { 24 }$
(2) $\frac { 1 } { 20 }$
(3) $\frac { 1 } { 16 }$
(4) $\frac { 1 } { 12 }$
(5) $\frac { 1 } { 8 }$

On the coordinate plane, let P $( t , \sin t ) ( 0 < t < \pi )$ be a point on the curve $y = \sin x$. Let circle $C$ be centered at P and tangent to the $x$-axis. Let Q be the point where circle $C$ is tangent to the $x$-axis, and let R be the point where circle $C$ meets segment OP.
If $\lim _ { t \rightarrow 0 + } \frac { \overline { \mathrm { OQ } } } { \overline { \mathrm { OR } } } = a + b \sqrt { 2 }$, find the value of $a + b$. (Here, O is the origin, and $a , b$ are integers.) [3 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

For what values of $a$ does $\displaystyle\lim_{x \to 0} \frac{\sin^a x}{x} = 0$?
(A) $a \geq 1$ (B) $a > 1$ (C) $a \leq 1$ (D) All real $a$

$\lim _ { x \rightarrow 0 } \left( \frac { x - \sin x } { x } \right) \sin \left( \frac { 1 } { x } \right)$
(1) equals 1
(2) equals 0
(3) does not exist
(4) equals $-1$

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

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

If $\lim _ { x \rightarrow 0 } \frac { a e ^ { x } - b \cos x + c e ^ { - x } } { x \sin x } = 2$, then $a + b + c$ is equal to

$$\lim _ { x \rightarrow 0 } \frac { x + \arcsin x } { \sin 2 x }$$
What is the value of this limit?
A) 0
B) 1
C) $\frac { 2 } { 3 }$
D) $\frac { 4 } { 3 }$
E) $\frac { 1 } { 6 }$

$$\lim _ { x \rightarrow \infty } \left( \sqrt { x ^ { 2 } + 2 x + 1 } - \sqrt { x ^ { 2 } + 1 } \right)$$
What is the value of this limit?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 5 } { 2 }$
D) 1
E) 2

$$\begin{aligned} & f ( x ) = 2 x - 1 \\ & g ( x ) = \frac { x } { 2 } - \frac { 1 } { x } \end{aligned}$$
Given this, what is the value of $\lim _ { x \rightarrow 2 } \frac { f ( g ( x ) ) } { x - 2 }$?
A) 0
B) 1
C) 3
D) $\frac { 1 } { 2 }$
E) $\frac { 3 } { 2 }$

Let m, n be non-zero real numbers. The function
$$f ( x ) = \frac { x } { n } \sin \left( \frac { m } { x } \right)$$
has a horizontal asymptote at $y = 2$. Given this, which of the following is the relationship between m and n?
A) $m = n$
B) $m = n + 2$
C) $m = 2 n$
D) $m = 3 n$
E) $2 m = 3 n$

$$f ( x ) = \left\{ \begin{array} { c c } \frac { a x } { x + 2 b } \cdot \cot x & , x \neq 0 \\ 2 & , x = 0 \end{array} \right.$$
The function is continuous at the point $x = 0$. Accordingly, what is the ratio $\frac { a } { b }$?
A) 1
B) 2
C) 4
D) $\frac { 1 } { 3 }$
E) $\frac { 1 } { 6 }$

$\lim _ { x \rightarrow \pi } \frac { x ^ { 2 } \cdot \sin ( \pi - x ) + \pi ^ { 2 } \cdot \sin ( x - \pi ) } { ( x - \pi ) ^ { 2 } }$\ What is the value of this limit?\ A) $- 2 \pi$\ B) $- \pi$\ C) $\pi$\ D) $2 \pi$\ E) $3 \pi$