Poles of the resolvent of a matrix

Question

Consider a linear map $A$ on a finite dimensional complex linear space $V$. Its resolvent operator is $R(x)=(x\mathbb{I}-A)^{-1}$. $R(\lambda)$ does not exist if and only if $x=\lambda$ is an eigenvalue of $A$. Now, the problem is to prove that there is some $n\leq \dim V$ such that the limit $$ \lim_{x\to\lambda}(x-\lambda)^n R(x) $$ does exist. Of course this can be done by using the Cramer's rule for the inverse of $(x\mathbb{I}-A)$. However, is it not possible to prove this from the definition of $R(x)$ and without introducing determinants?