Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$ 
Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$

Going from the epsilon delta definition we get: 
$$\forall x,y>1,\text{WLOG}:x>y \ ,\ \forall\epsilon>0,\exists\delta>0 \ s.t. \ |x-y|<\delta\rightarrow |x+\frac{x}{x+1}-y-\frac{y}{y+1}|<\epsilon$$
But I'm not really sure on how to continue from here.
Alternatively: if $\lim\limits_{x\to \infty}f(x)$ exists on $[0,\infty)$ then the function is uniformly continuous, well, it's easy to see here that $\lim\limits_{x\to \infty}f(x)=\infty$ but is that enough ?

Take a smaller bite.  Note that a sum of uniformly continuous functions is uniformly continuous.  We know $x\mapsto x$ is uniformly continuous, so we just deal with $x/(x+1)$.
Also, notice that 
$${x\over x + 1 } = 1 - {1\over x + 1}.$$
Now the job is to show that $x\mapsto 1/(x + 1)$ is uniformly continuous.  Can you do the rest?
