Let $g$ be a continuously differentiable function with $g ( 1 ) = 6$ and $g ^ { \prime } ( 1 ) = 3$. What is $\lim _ { x \rightarrow 1 } \frac { \int _ { 1 } ^ { x } g ( t ) d t } { g ( x ) - 6 }$ ?
(A) 0
(B) $\frac { 1 } { 2 }$
(C) 1
(D) 2
(E) The limit does not exist.