I was having this problem with my bike recently. While the FD can switch without an issue between the second and third (largest) ring, it can't get the chain to the first ring (smallest). I recalibrated the FD but the problem remains as it is. I noticed though that no matter how much I unscrewed the "L" screw, for the known reasons, the FD didn't reached the new limit but only when I gave it a slight push with my fingers. Is there anything I can do to make the FD reach the "L" limit screw? Btw the FD is an Shimano FD-TZ500. Thanks!
Either the inboard travel of the derailleur cage is limited by the adjustment of the shifter cable rather than the limit screw, or there is something impeding the derailleur movement.
if the shift cable is tight and you can move the derailleur by adjusting the barrel adjuster then the derailleur is not up against the limit screw.
If the cable is loose, or you can manually push the derailleur up against the limit screw there is something impeding the derailleur movement. Excess friction or binding of the cable in the housing is a common cause of this.
You can pull the cable out of the housing, clean and lubricate with a small amount of chain lube, or just replace the cable and housing as they are not very expensive.