$\lim _ { x \rightarrow \pi } \frac { \cos x + \sin ( 2 x ) + 1 } { x ^ { 2 } - \pi ^ { 2 } }$ is
(A) $\frac { 1 } { 2 \pi }$
(B) $\frac { 1 } { \pi }$
(C) 1
(D) nonexistent

Two real numbers $a$ and $b$ satisfy $\lim _ { x \rightarrow 2 } \frac { \sqrt { x ^ { 2 } + a } - b } { x - 2 } = \frac { 2 } { 5 }$. Find the value of $a + b$. [3 points]

For the function $f ( x ) = x ^ { 4 } + 4 x ^ { 2 } + 1$, find the value of $$\lim _ { h \rightarrow 0 } \frac { f ( 1 + 2 h ) - f ( 1 ) } { h }$$. [3 points]

For a function $f ( x )$ differentiable on the set of all real numbers, let the function $g ( x )$ be defined as $$g ( x ) = \frac { f ( x ) } { e ^ { x - 2 } }$$ If $\lim _ { x \rightarrow 2 } \frac { f ( x ) - 3 } { x - 2 } = 5$, what is the value of $g ^ { \prime } ( 2 )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Let $f$ be a differentiable function with $f(3) \neq 0$. Evaluate $\displaystyle\lim_{x \to \infty} \left(\frac{f(3 + 1/x)}{f(3)}\right)^x$.
(A) $e^{f'(3)/f(3)}$ (B) $e^{f(3)}$ (C) $e^{f'(3)}$ (D) 1

If $f ( x ) = \left| \begin{array} { c c c } \cos x & x & 1 \\ 2 \sin x & x ^ { 2 } & 2 x \\ \tan x & x & 1 \end{array} \right|$, then $\lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x }$
(1) does not exist
(2) exists and is equal to $-2$
(3) exists and is equal to 0
(4) exists and is equal to 2

$f ( x ) = \left| \begin{array} { c c c } \cos x & x & 1 \\ 2 \sin x & x ^ { 2 } & 2 x \\ \tan x & x & 1 \end{array} \right|$, then $\lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x }$
(1) Exists and is equal to - 2
(2) Does not exist
(3) Exist and is equal to 0
(4) Exists and is equal to 2

Let $f : R \rightarrow R$ be a differentiable function satisfying $f ^ { \prime } ( 3 ) + f ^ { \prime } ( 2 ) = 0$. Then $\lim _ { x \rightarrow 0 } \left( \frac { 1 + f ( 3 + x ) - f ( 3 ) } { 1 + f ( 2 - x ) - f ( 2 ) } \right)$ is equal to
(1) 1
(2) e
(3) $e ^ { 2 }$
(4) $e ^ { - 1 }$

Let $f(x)$ be a polynomial function such that $f(x) + f ^ { \prime } (x) + f ^ { \prime \prime } (x) = x ^ { 5 } + 64$. Then, the value of $\lim _ { x \rightarrow 1 } \frac { f(x) } { x - 1 }$ is equal to
(1) $- 15$
(2) $15$
(3) $- 60$
(4) $60$

Let $f : ( - \infty , \infty ) - \{ 0 \} \rightarrow \mathbb { R }$ be a differentiable function such that $f ^ { \prime } ( 1 ) = \lim _ { a \rightarrow \infty } a ^ { 2 } f \left( \frac { 1 } { a } \right)$. Then $\lim _ { a \rightarrow \infty } \frac { a ( a + 1 ) } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { a } \right) + a ^ { 2 } - 2 \log _ { e } a$ is equal to
(1) $\frac { 3 } { 2 } + \frac { \pi } { 4 }$
(2) $\frac { 3 } { 4 } + \frac { \pi } { 8 }$
(3) $\frac { 3 } { 8 } + \frac { \pi } { 4 }$
(4) $\frac { 5 } { 2 } + \frac { \pi } { 8 }$