About Abel's Theorem about power series.

Question

I am reading "A First Course in Analysis Vol.1" by Sin Hitotumatu.

There is the following theorem in this book.

Theorem:

1. $f(z) := \sum_{n=0}^\infty a_n z^n$.
2. The radius of convergence of $f(z)$ is $1$.
3. $\sum_{n=0}^\infty a_n = f(1)$ converges.
4. $\alpha \in (0, \frac{\pi}{2})$.
5. $A := \{z \in \mathbb{C} | |z| < 1\} \cap \{z \in \mathbb{C} | -\alpha \leq \arg(1-z) \leq \alpha\}$.

Then, $\lim_{z \in A, z \to 1} f(z) = f(1)$.

Let $B := \{z \in \mathbb{C} | |z| < 1\}$.

Please tell me an example such that $\lim_{z \in B, z \to 1} f(z) \ne f(1)$.