Proving that $\nabla u$ is not bounded near $x=0$ for the Dirichlet problem for the half-space So I am trying to solve a problem from Evan's book, which asks to prove that $\nabla u$ is not bounded near $x=0$, where $u$ is the solution of 
$$\begin{cases} \Delta u = 0 & \text{in } \mathbb{R}_+^n\\
u = g & \text{on } \partial\mathbb{R}_+^n \end{cases}$$
where $\mathbb{R}_+^n = \{(x_1,\ldots,x_n)\in\mathbb{R}^n: x_n>0\}$ and $g$ is bounded and $g(x) = |x|$ for $|x|\leq 1$.
I found this solution in http://users.math.msu.edu/users/kpromisl/math849/HW2A.pdf which I was able to follow because it is very clear, but one thing stroke me as odd. They separate $\frac{u(\lambda e_n) - u(0)}{\lambda}$ as the addition of two integrals, and then argue that one of those integrals is convergent and therefore finite for any $\lambda$, while the other is divergent for any $\lambda$.
But wouldn't that mean that $u(\lambda e_n) = \infty$ for any $\lambda > 0$? But how can that be if $u$ is supposed to be well-defined in $\mathbb{R}_+^n$, which obviously includes any $x = \lambda e_n$ for $\lambda > 0$?

I don't understand why they claim that $J$ is divergent. But what seems to be true is that $J$ tends to $+\infty$ when $\lambda \to 0$. And we can conclude from that.
EDIT: to be more precise $r^{n-1}/(\lambda^2+r^2)^{n/2}$ is continuous on $[0,1]$ (because $\lambda>0$) so $J$ is convergent. But when $\lambda \to 0$ then $r^{n-1}/(\lambda^2+r^2)^{n/2} \sim 1/r$ so we get an divergent integral.
