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

10. $\lim _ { h \rightarrow 0 } \frac { \sin ( x + h ) - \sin x } { h }$ is
(A) 0
(B) 1
(C) $\quad \sin x$
(D) $\cos x$
(E) nonexistent

For a differentiable function $f$, let $f ^ { * }$ be the function defined by $f ^ { * } ( x ) = \lim _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x - h ) } { h }$. (a) Determine $f ^ { * } ( x )$ for $f ( x ) = x ^ { 2 } + x$. (b) Determine $f ^ { * } ( x )$ for $f ( x ) = \cos x$. (c) Write an equation that expresses the relationship between the functions $f ^ { * }$ and $f ^ { \prime }$, where $f ^ { \prime }$ denotes the usual derivative of $f$.

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]

$\lim _ { x \rightarrow a } \frac { 2 ^ { x } - 1 } { 3 \sin ( x - a ) } = b \ln 2$ 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) $\frac { 1 } { 2 }$

A polynomial function $f ( x )$ and two natural numbers $m , n$ satisfy $$\begin{array} { l l } \lim _ { x \rightarrow \infty } \frac { f ( x ) } { x ^ { m } } = 1 , & \lim _ { x \rightarrow \infty } \frac { f ^ { \prime } ( x ) } { x ^ { m - 1 } } = a \\ \lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { n } } = b , & \lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x ^ { n - 1 } } = 9 \end{array}$$ Which of the following are correct? Select all that apply from . (where $a , b$ are real numbers.) [4 points]
ㄱ. $m \geqq n$ ㄴ. $a b \geqq 9$ ㄷ. If $f ( x )$ is a cubic function, then $a m = b n$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a polynomial function $f ( x )$, if $\lim _ { x \rightarrow 2 } \frac { f ( x + 1 ) - 8 } { x ^ { 2 } - 4 } = 5$, find the value of $f ( 3 ) + f ^ { \prime } ( 3 )$. [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

111- If $f(x) = \sqrt{x^2 - |x| + |x|}$, then $\displaystyle\lim_{h \to 0^+} \dfrac{f(1+h)-f(1)}{h}$ is which of the following?
(1) $\dfrac{1}{2}$ (2) $\dfrac{5}{4}$ (3) $\dfrac{3}{2}$ (4) $\dfrac{5}{2}$

17 -- If $f(x) = \left(\dfrac{-1+\sin x}{1+\sin x}\right)^2$ and $f(x) = xg(x)+1$, then $\displaystyle\lim_{x \to 0} g(x)$ equals:
(1) $4$ (2) $2$ (3) $-4$ (4) $-2$

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

23. Let $f : R \rightarrow R$ be such that $f ( 1 ) = 3$, and $f ^ { \prime } ( 1 ) = 6$ Then $\lim _ { x \rightarrow 0 } ( f ( 1 + x ) / f ( 1 ) ) ^ { 1 / x }$ equals
(A) 1
(B) $\quad \mathrm { e } ^ { 1 / 2 }$
(C) $\quad \mathrm { e } ^ { 2 }$
(D) $\quad \mathrm { e } ^ { 3 }$

13. $\lim _ { ( h \rightarrow 0 ) } \left( \mathrm { f } \left( 2 \mathrm {~h} + 2 + \mathrm { h } ^ { 2 } \right) - \mathrm { f } ( 2 ) \right) / \left( \mathrm { f } \left( \mathrm { h } - \mathrm { h } ^ { 2 } + 1 \right) - \mathrm { f } ( 1 ) \right)$, given that $f ^ { \prime } ( 2 ) = 6$ and $f ^ { \prime } ( 1 ) = 4$ :
(a) does not exists
(b) is equal to $- 3 / 2$
(c) is equal to $3 / 2$
(d) is equal to 3

7. If $f : [ - 1,1 ] \rightarrow R$ and $f ^ { \prime } ( 0 ) = \lim _ { n \rightarrow \infty } n f \left( \frac { 1 } { n } \right)$ and $f ( 0 ) = 0$. Find the value of $\lim _ { n \rightarrow \infty } \frac { 2 } { \pi } ( n + 1 ) \cos ^ { - 1 } \left( \frac { 1 } { n } \right) - n$.
Given that $0 < \left| \lim _ { \mathrm { n } \rightarrow \infty } \cos ^ { - 1 } \left( \frac { 1 } { \mathrm { n } } \right) \right| < \frac { \pi } { 2 }$. Sol. $\lim _ { n \rightarrow \infty } \frac { 2 } { \pi } ( n + 1 ) \cos ^ { - 1 } \frac { 1 } { n } - n = \lim _ { n \rightarrow \infty } n \left[ \frac { 2 } { \pi } \left( 1 + \frac { 1 } { n } \right) \cos ^ { - 1 } \frac { 1 } { n } - 1 \right]$ $= \lim _ { n \rightarrow \infty } n f \left( \frac { 1 } { n } \right) = f ^ { \prime } ( 0 )$ where $f ( x ) = \frac { 2 } { \pi } ( 1 + x ) \cos ^ { - 1 } x - 1$. Clearly, $\mathrm { f } ( 0 ) = 0$.
Now, $f ^ { \prime } ( x ) = \frac { 2 } { \pi } \left[ ( 1 + x ) \frac { - 1 } { \sqrt { 1 - x ^ { 2 } } } + \cos ^ { - 1 } x \right]$ $\Rightarrow \mathrm { f } ^ { \prime } ( 0 ) = \frac { 2 } { \pi } \left[ - 1 + \frac { \pi } { 2 } \right] = \frac { 2 } { \pi } \left[ \frac { \pi - 2 } { 2 } \right] = 1 - \frac { 2 } { \pi }$.

7. If $f : [ - 1,1 ] \rightarrow R$ and $f ^ { \prime } ( 0 ) = \lim _ { n \rightarrow \infty } n f \left( \frac { 1 } { n } \right)$ and $f ( 0 ) = 0$. Find the value of $\lim _ { n \rightarrow \infty } \frac { 2 } { \pi } ( n + 1 ) \cos ^ { - 1 } \left( \frac { 1 } { n } \right) - n$.
Given that $0 < \left| \lim _ { \mathrm { n } \rightarrow \infty } \cos ^ { - 1 } \left( \frac { 1 } { \mathrm { n } } \right) \right| < \frac { \pi } { 2 }$. Sol. $\lim _ { n \rightarrow \infty } \frac { 2 } { \pi } ( n + 1 ) \cos ^ { - 1 } \frac { 1 } { n } - n = \lim _ { n \rightarrow \infty } n \left[ \frac { 2 } { \pi } \left( 1 + \frac { 1 } { n } \right) \cos ^ { - 1 } \frac { 1 } { n } - 1 \right]$ $= \lim _ { n \rightarrow \infty } n f \left( \frac { 1 } { n } \right) = f ^ { \prime } ( 0 )$ where $f ( x ) = \frac { 2 } { \pi } ( 1 + x ) \cos ^ { - 1 } x - 1$. Clearly, $\mathrm { f } ( 0 ) = 0$.
Now, $f ^ { \prime } ( x ) = \frac { 2 } { \pi } \left[ ( 1 + x ) \frac { - 1 } { \sqrt { 1 - x ^ { 2 } } } + \cos ^ { - 1 } x \right]$ $\Rightarrow \mathrm { f } ^ { \prime } ( 0 ) = \frac { 2 } { \pi } \left[ - 1 + \frac { \pi } { 2 } \right] = \frac { 2 } { \pi } \left[ \frac { \pi - 2 } { 2 } \right] = 1 - \frac { 2 } { \pi }$.

15. If $f ( x - y ) = f ( x ) \cdot g ( y ) - f ( y ) \cdot g ( x )$ and $g ( x - y ) = g ( x ) \cdot g ( y ) + f ( x ) \cdot f ( y )$ for all $x$, $y \hat { I } R$. If right hand derivative at $x = 0$ exists for $f ( x )$. Find derivative of $g ( x )$ at $x = 0$.

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) $\frac { 5 } { 2 } + \frac { \pi } { 8 }$

Q85. Let $f$ be a differentiable function in the interval $( 0 , \infty )$ such that $f ( 1 ) = 1$ and $\lim _ { t \rightarrow x } \frac { t ^ { 2 } f ( x ) - x ^ { 2 } f ( t ) } { t - x } = 1$ for each $x > 0$. Then $2 f ( 2 ) + 3 f ( 3 )$ is equal to $\_\_\_\_$

Q68. 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) $\frac { 5 } { 2 } + \frac { \pi } { 8 }$

Let $\mathrm { p } ( \mathrm { x } )$ be a differentiable function such that $\mathrm { p } ( 1 ) = 2$.
If $\operatorname { Lim } _ { t \rightarrow x } \left( \frac { t ^ { 2 } p ( x ) - x ^ { 2 } p ( t ) } { x - t } \right) = 3$, then the value of $2 p ( 2 )$.