At first, to identify the certain user or group limits you have to do the following:
root@ubuntu:~# sudo -u mysql bash
mysql@ubuntu:~$ ulimit -a
core file size (blocks, -c) 0
data seg size (kbytes, -d) unlimited
scheduling priority (-e) 0
file size (blocks, -f) unlimited
pending signals (-i) 71680
max locked memory (kbytes, -l) 32
max memory size (kbytes, -m) unlimited
open files (-n) 1024
pipe size (512 bytes, -p) 8
POSIX message queues (bytes, -q) 819200
real-time priority (-r) 0
stack size (kbytes, -s) 8192
cpu time (seconds, -t) unlimited
max user processes (-u) 71680
virtual memory (kbytes, -v) unlimited
file locks (-x) unlimited
mysql@ubuntu:~$
The important line is:
open files (-n) 1024
As you can see you OS vendor ships this version with the basic Linux configuration which is 1024 files per process.
This is obviously not enough for a busy MySQL installation.
Now, to fix this you have to modify the following file:
/etc/security/limits.conf
mysql soft nofile 24000
mysql hard nofile 32000