How can I programmatically determine what the resource limits pertaining to the currently-running process are?

Maybe getrlimit(2) — at least that's the system call used under the hood of the shell builtin ulimit.

There's also the module BSD::Resource (a wrapper around the related system calls getrlimit, setrlimit, getrusage, etc.):

use BSD::Resource;

my $rlimits = get_rlimits();
for my $rl (sort keys %$rlimits) {
    printf "%-15s : %s\n", $rl, getrlimit($rlimits->{$rl});
}

__END__
RLIMIT_AS       : 3412459520
RLIMIT_CORE     : 0
RLIMIT_CPU      : -1
RLIMIT_DATA     : -1
RLIMIT_FSIZE    : -1
RLIMIT_LOCKS    : -1
RLIMIT_MEMLOCK  : 32768
RLIMIT_NOFILE   : 1024
RLIMIT_NPROC    : 16383
RLIMIT_OFILE    : 1024
RLIMIT_OPEN_MAX : 1024
RLIMIT_RSS      : 1794068480
RLIMIT_STACK    : 8388608
RLIMIT_VMEM     : 3412459520
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