Turn on a stovetop burner and place poblano chile over open flame; carefully turn to char on all sides. Place poblano in a plastic or paper bag; close bag and set aside 10 minutes to steam. Rub poblano skin off with hands under cold running water; core, seed and dice poblano.
Place pork in a 5-quart slow cooker. Add broth, sweet potato, onion, poblano chile, tomatoes, cumin, salt and pepper; stir. Cover and cook on LOW setting for 6 to 7 hours. Add lime juice and stir well.