We consider the linear growth and fragmentation equation with general coefficients. Under suitable conditions, the first eigenvalue represents the asymptotic growth rate of solutions, also called \emph{fitness} or \emph{Malthus coefficient} in population dynamics ; it is of crucial importance to understand the long-time behaviour of the population. We investigate the dependency of the dominant eigenvalue and the corresponding eigenvector on the transport and fragmentation coefficients. We show how it behaves asymptotically as transport dominates fragmentation or \emph{vice versa}. For this purpose we perform suitable blow-up analysis of the eigenvalue problem in the limit of small/large growth coefficient (resp. fragmentation coefficient). We exhibit possible non-monotonic dependency on the parameters, conversely to what would have been conjectured on the basis of some simple cases.