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The external display's max. resolution is 2560 x 1440, but the maximum resolution I can select on the display settings is 2048 x 1152.
What can I do to use the display in its max. resolution?
System: elementary OS 0.4.1 Loki
Display driver: Intel Corporation 3rd Gen Core processor Graphics Controller (rev 09)
Based on answer
you can use gtf height width refresh_rate and get an new modeline as response and then you can create new mode with the output of the first command xrandr --newmode modeline, after that you need to ensure in which output you want to set up the modeline, you can check this with xrandr -q after get the response add the newmode to the desired output xrandr --addmode OUTPUT "MODELINE" and apply the new modeline xrandr --output OUTPUT --mode MODELINE.
Please open terminal pressing Ctrl+Alt+T and execute xrandr -q to see the connected displays and their current resolution.
Then execute xrandr --output <monitor_name> --mode <resolution> to change resolution. For example, xrandr --output eDP-2-1 --mode 2560x1440
xrandr output. Since I posted the issue I've tried all sorts of possible solutions, the closest one I got is 2560x1440 resolution at 30Hz 2560x1440 or 2560x1600 via HDMI which is not acceptable. I think it is my display driver that cannot handle the resolution.
xrandr -qthis return to you the min and the max resolutions