Cook the tagliatelle in plenty of salted water according to the instructions on the package, until it is firm to the bite.
Peel the asparagus, cut off the ends and cook the asparagus in plenty of salted water. Lift the asparagus stalks out of the water, allow to drain. Cut the asparagus into 3-4 cm pieces and fry lightly in a pan with 2 tablespoons of sunflower oil for about 5 minutes.
Wash the lime with hot water, dab and rub the peel. Then halve and squeeze the lime. Peel the garlic clove, cut in half into thin slices, roughly chop. Mix the lime juice zest and 4 tablespoons of sunflower oil in a bowl. Fold in the garlic, add salt, pepper and Season the sugar to taste.
Remove the tagliatelle from the boiling water and drain well. Sort out the arugula, clean, wash and shake dry. Lift the tagliatelle out of the water with a ladle and drain well. Add the arugula to the asparagus and fold in slightly. Add the tagliatelle with the lime sauce to the arugula -Mix asparagus. Season to taste with salt and pepper.
Arrange the asparagus and rocket tagliatelle on plates and sprinkle with the Parmesan.