Slice lime into quarters, place them pulp-side-up into a squat old-fashioned glass and add simple syrup. Muddle lime pieces with a wooden pestle until most of the juice is free of the lime. Leave the lime quarters in the glass. Add the cachaça and fill the glass with crushed ice to just below the lip. Shake and serve immediately.
To make simple syrup: Simmer equal parts sugar and water over medium heat until sugar dissolves. Remove from heat and cool prior to use.