Answer all the following questions. I. Find the following limit value: $$\lim _ { x \rightarrow 0 } \frac { b ^ { x } - c ^ { x } } { a x } \quad ( a , b , c > 0 )$$ II. Find the general solutions of the following differential equations. $$\begin{aligned}
& \text { 1. } \frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { y } { x } = \log x \quad ( x > 0 ) \\
& \text { 2. } \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = 2 x ^ { 2 } + 2 x
\end{aligned}$$ III. Let $a _ { n }$ be defined by $$a _ { n } = \frac { n ! } { n ^ { n + \frac { 1 } { 2 } } e ^ { - n } }$$ where $n$ is a positive integer and $e$ is the base of natural logarithm. Find $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { a _ { n + 1 } }$. Note that the function $y = x ^ { - 1 } ( x > 0 )$ is convex downward.
Answer all the following questions.
I. Find the following limit value:
$$\lim _ { x \rightarrow 0 } \frac { b ^ { x } - c ^ { x } } { a x } \quad ( a , b , c > 0 )$$
II. Find the general solutions of the following differential equations.
$$\begin{aligned}
& \text { 1. } \frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { y } { x } = \log x \quad ( x > 0 ) \\
& \text { 2. } \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = 2 x ^ { 2 } + 2 x
\end{aligned}$$
III. Let $a _ { n }$ be defined by
$$a _ { n } = \frac { n ! } { n ^ { n + \frac { 1 } { 2 } } e ^ { - n } }$$
where $n$ is a positive integer and $e$ is the base of natural logarithm.
Find $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { a _ { n + 1 } }$.
Note that the function $y = x ^ { - 1 } ( x > 0 )$ is convex downward.