Note that this is not a math question, but a language one. Nonetheless it requires some understanding of math and I thus hope it is appropriate to be asked here. Also be warned of non rigorous mathematics
Assume we want to calculate the deflection of a beam, which requires solving
$$ -T_0\frac{d^2u}{dx^2}+\rho g\sqrt{1+(\frac{du}{dx})^2}=0. $$ To make life easier we assume that $\frac{du}{dx}\ll 1$ which yields a linear equation which is trivial to solve. How can one more elegantly phrase
In order to check that the assumptions that were made are at least not too obviously violated, we compute $\dots$ and find $\frac{du}{dx}\ll1$ (where $u$ is the solution of the linear equation).
I can't just say "in order to check that our assumptions are indeed satisfied" because that can plainly not be concluded from looking at the solution of the linear equation. Notwithstanding this logical gap, no one will take you seriously if you do not have such an "assumption check". Another alternative phrasing would be
We calculate $\frac{du}{dx}\dots \ll 1$, which does not contradict our assumption.
but I find this little informative, especially if it comes after multiple pages of computations.