$\hat{J}_+ |j, j\rangle = \hbar\sqrt{(j-j)(j+j+1)} |j, j+1\rangle = 0$ ✓ (top of ladder) - $\hat{J}_- |j, -j\rangle = \hbar\sqrt{(j-j)(-j+j+1)} |j, -j-1\rangle = 0$ ✓ (bottom of ladder) - $\hat{J}_+ |1/2, -1/2\rangle = \hbar\sqrt{(1)(1)} |1/2, 1/2\rangle = \hbar |1/2, 1/2\rangle$ ✓