The documentation on this is fuzzy, but it turns out you can do this with shell builtins: 

# Bash
if [[ -L $symlink && ! -e $symlink ]]; then
  echo "$symlink is dangling"
fi

The -e test (like all other tests besides -L or -h) follows the symbolic link, and therefore it checks on the thing pointed to, not on the link itself. The -L test does not follow the symlink, so it's checking on the link itself. Together, they can indicate the presence of a dangling symlink.

POSIX has these same tests, with the same semantics, so it can also be done with the [ command: 

# POSIX
if [ -L "$symlink" ] && [ ! -e "$symlink" ]; then
  echo "$symlink is dangling"
fi