Scale-free networks arise from power-law degree distributions. Due to the finite size of real-world networks, the power law inevitably has a cutoff at some maximum degree $Δ$. We investigate the relative size of the giant component $S$ in the large-network limit. We show that $S$ as a function of $Δ$ increases fast when $Δ$ is just large enough for the giant component to exist, but increases ever more slowly when $Δ$ increases further. This makes that while the degree distribution converges to a pure power law when $Δ\to\infty$, $S$ approaches its limiting value at a slow pace. The convergence rate also depends on the power-law exponent $τ$ of the degree distribution. The worst rate of convergence is found to be for the case $τ\approx2$, which concerns many of the real-world networks reported in the literature.