\displaystyle \lim_{(x_0, \frac{1}{\frac{\pi}{2} + 2 n \pi}) \to (x_0, 0)} f(x, y) = \lim_{n \to +\infty} f \left( x_0, \frac{1}{\frac{\pi}{2} + 2 n \pi} \right) = x_0 \sin \frac{1}{x_0}