In Section 7.2.1 of Bergman's Fundamentals of Heat and Mass Transfer, there is a derivation of the Blasius equation $$ 2 \frac{\mathrm d^3 f}{\mathrm d \eta^3} + f \frac{\mathrm d^2 f}{\mathrm d \eta^2} = 0$$ through the definitions $$ \eta = y \sqrt{\frac{u_\infty}{\nu x}} \quad \text{and} \quad f(\eta) = \frac{\psi}{u_\infty \sqrt{\frac{\nu x}{u_\infty}}},$$ where $\psi$ is the velocity streamfunction satisfying $ u = \frac{\partial \psi}{\partial y} $ and $ v = - \frac{\partial \psi}{\partial x}$.
I have derived for myself the transformation from the momentum equation $$ u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2} $$ to the Blasius equation, which was mainly a lot of chain rule and algebraic manipulation.
I understand that the no-slip and freestream BCs are $$ u(x, 0) = 0, \quad v(x, 0) = 0, \quad u(x, \infty) = u_\infty.$$
Since the first two no-slip BCs are at $y = 0$, plugging this into the definition of $\eta$ yields that $\eta = 0$ at the wall. Similarly for the freestream, as $y \to \infty$, $\eta \to \infty$. But how do I derive the listed BCs? Unfortunately, Bergman's textbook simply states them without explanation. $$ f(\eta = 0) = 0, \quad \left.\frac{\mathrm df}{\mathrm d\eta}\right|_{\eta = 0} = 0, \quad \left.\frac{\mathrm df}{\mathrm d \eta}\right|_{\eta \to \infty} = 1.$$
