Let $A$ and $B(t)$ dependent on parameter $t$ be $n\times n$ matrices. $B(t)$ is positive definite for all $t>0$. Also, let $C$ be an $n \times m$ ($m<n$) matrix. We know that $\lim_{t \to 0}\frac{1}{t^2}B(t) = B_0$, where $B_0$ is positive semi-definite and $B_0C=0$. $A$ is symmetric and positive semi-definite such that $A=X^{\top}B_0X$, where $X$ is an invertible $n \times n$ matrix. Show that $\lim_{t \to 0}(A+B(t))^{-1}B(t)C=0$?
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We assume $C\not= 0$. Since $m<n$, $C^T$ is not one to one and there is $x\in \mathbb{R}^n\setminus\{0\}$ s.t. $C^Tx=0$; then let $B_0=xx^T$ and $B(t)=t^2B_0+tI_n$.
We choose $A=0$ and the considered expression is $C$, that does not tend to $0$.