Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Let us fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$.

Now let $\Lambda_{\xi, q}$, with $\xi \in C$, be a flat family of linear spaces parametrized by $C$. Assume that for a general $\xi\in C$ the intertsection multiplicity of $\Lambda_{\xi, q}$ and $X$ at $\xi$ is greater or equal than $m_1$, and the intertsection multiplicity of $\Lambda_{\xi, q}$ and $X$ at $q$ is greater or equal than $m_2$.

Finally, let $\Lambda_{q, q}$ be the limit linear space of the family $\Lambda_{\xi, q}$ when $\xi\mapsto q$ along $C$.

May we then conclude that the intertsection multiplicity of $\Lambda_{q, q}$ and $X$ at $q$ is greater or equal than $m_1+m_2$ ?