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