# The decoupling limit of the linear sigma model

I'm reading Schwartz's Quantum field theory and the standard model. I'm stuck at page 566. He says that when we take the decoupling limit, that's keeping $F_{\pi}$ fixed while sending $m$ and $\lambda$ to infinity, projects out the non-linear sigma model. Now, I don't understand what all this means. Doing this will take the coefficients of all terms in the potential to infinity while not affecting the kinetic terms.

So, How did he get:

$$L=(1/2) (\partial_{\mu}\pi)^2$$ after taking the decoupling limit?

Substitute $F_\pi$ = $\frac{2m}{\sqrt{\lambda}}$ in (28.12 ) so the second term in between the parenthesis that's multiplied by $1/F^2_\pi$ becomes $\frac{1}{2} (\partial\pi)^2 + other \; terms$. Apply L'Hospital's rule and ignore all the $\infty$'s. Finally the Lagrangian becomes ($decoupled \: \pi$ & $\sigma$ fields): $\frac{1}{2} (\partial\pi)^2 + \frac{1}{2} (\partial\sigma)^2$