1. Let $f(x)$ be the function given by:
\[
f(x)=\frac{1}{x+1}
\]
Find $f(a), f(a+h)$, and the difference quotient
\[
\frac{f(a+h)-f(a)}{h}
\]
Simplify your answers completely.

Step 1 ：Let's start by finding the values of $f(a)$ and $f(a+h)$ by substituting $a$ and $a+h$ into the function $f(x)$.

Step 2 ：For $f(a)$, we substitute $a$ into $f(x)$ to get $f(a) = \frac{1}{a+1}$.

Step 3 ：For $f(a+h)$, we substitute $a+h$ into $f(x)$ to get $f(a+h) = \frac{1}{a+h+1}$.

Step 4 ：Next, we calculate the difference quotient by substituting $f(a+h)$ and $f(a)$ into the difference quotient formula $\frac{f(a+h)-f(a)}{h}$.

Step 5 ：Substituting the values we found for $f(a+h)$ and $f(a)$ into the difference quotient formula, we get $\frac{\frac{1}{a+h+1} - \frac{1}{a+1}}{h}$.

Step 6 ：Simplifying this expression, we find that the difference quotient is $\frac{-1}{(a+1)(a+h+1)}$.

Step 7 ：\(\boxed{f(a) = \frac{1}{a+1}, f(a+h) = \frac{1}{a+h+1}, \text{ and the difference quotient is } \frac{-1}{(a+1)(a+h+1)}}\)