By definition:
Suppose $lim x_n = x_0$ $lim f(x_n)$ = $lim \sqrt{2x_n+1}$ = $\sqrt{2[lim (x_n)] +1}$ = $\sqrt{2x_0 +1}$ = $f(x_0)$
By epsilon-delta property:
Let $\epsilon > 0$. We want $|f(x) - f(x_0)| < \epsilon$, while $|x-x_0|<\delta$.
$|f(x)-f(x_0)|$ = $|\sqrt{2x+1}-\sqrt{2x_0+1}|$ = $|\sqrt{2x+1}-\sqrt{2(-0.5)+1}$=$\sqrt{2x+1}<\epsilon$
I think this is a good start (but correct me if it's not), but I am clueless as to what to do from here.
Just looking for tips and corrections if need be. Please do not solve for me.