In a resealable plastic bag, combine the lime juice, garlic, vinegar, oil, salt and pepper. Add the steak and shake gently to coat. Put the sealed bag in a shallow dish and refrigerate for 10 to 15 minutes.
Preheat a grill pan over medium-high heat. Lift the steak from the bag and let any excess marinade drip off. Discard the marinade. Grill the steak for 5 to 7 minutes on each side for medium-rare or until a thermometer inserted in the center registers 145 degrees F. Transfer the steak to a cutting board and let it rest for 5 to 10 minutes to reabsorb its juices. Thinly slice the steak across the grain.
In a medium bowl, toss the arugula with the vinaigrette. Divide the salad among 4 plates, top with the sliced steak, and serve.