This problem is from a set of exercises that I have. It states:
Let $u\in C(\overline{\mathbb{R}_+^2})$ be a bounded harmonic function in the upper half plane $\mathbb{R}_+^2$, satisfying $u(x,0) \to \pi$ as $x\to \infty$ and $u(x,0)\to 0$ as $x \to -\infty$. Compute the limit of $u(r\cos\theta, r\sin\theta)$ as $r\to\infty$ for each $0<\theta < \pi$.
I tried using the Poisson formula for the half plane but didn't manage to get anywhere. I appreciate any input!
Hint: Suppose $u(t,0)\to 0$ as $|t|\to \infty,$ with the other hypotheses unchanged. Let $v(z) = u(-1/z).$ Show $v$ is continuous at the origin, with value $0$ there. Show that this implies $\lim_{r\to \infty} u(re^{it})=0$ for all $t\in (0,\pi).$ Now to the original problem: There is an easy bounded harmonic function $u_0$ that equals $\pi$ on the positive real axis and $0$ on the negative real axis. Given $u$ as in your problem, apply the above to $u-u_0.$