I am attempting to solve a problem in the field of Economics, and for that purpose I have devised the following lemmas.
Lemma 1: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ has a nontrivial solution for $z$. Let $A(x)y(x)=b$, where $b>0$. Then $\lim_{x \rightarrow x^*} y(x)=\infty$, where $\infty$ is a vector where at least one entry is equal to infinity.
Lemma 2: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ has a nontrivial solution for $z$. Let $A(x)y(x)=b$, where $b>0$. Let $z$ have no zero entries. Then $\lim_{x→x^*}y(x)=\infty^{*}$, where $∞^*$ is a vector where all entries are equal to infinity.
I have look on many books to find these theorems or a perfected form of them, with no results. Can someone help?