Answer
$L(x)=\frac{1}{4}x+\frac{1}{4}$
Work Step by Step
Step 1. $f(x)=\frac{x}{x+1}$, $f'(x)=\frac{x+1-x}{(x+1)^2}=\frac{1}{(x+1)^2}$,
Step 2. Choose $x=1$, $f(1)=\frac{1}{2}$, $f'(1)=\frac{1}{(1+1)^2}=\frac{1}{4}$
Step 3. The linearization at $x=1$ is $L(x)=f(1)+f'(1)(x-0)=\frac{1}{2}+\frac{1}{4}(x-1)=\frac{1}{4}x+\frac{1}{4}$
