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$$

\lim_{r\to 1}P(r,\theta) = \begin{cases}

\infty, & \theta = 0\\

0, & \text{otherwise}

\end{cases}

$$

For the first piece, take the summation

$$

P(1,0) = \frac{1}{\pi}\left(\frac{1}{2} + \sum_{n = 1}^{\infty} 1^n\right).

$$

Then $\sum\limits_{n = 1}^{\infty} 1^n = \infty$.

Therefore, we have a positive number plus infinity which is infinity when $r\to 1$ and $\theta = 0$.

For the second piece, take the fractional representation of the Poisson kernel,

$$

P(1,\theta) = \frac{1}{2\pi}\frac{0}{2 - 2\cos\theta} = 0.

$$

Therefore, $P(r,\theta) = 0$ for all $\theta\neq 0$.

That is,

$$\lim_{r\to 1}P(r,\theta) = \begin{cases}

\infty, & \theta = 0\\

0, & \text{otherwise}

\end{cases}

$$