Cut the lime into 8 parts: first cut off the bottom and the top of the lime (only the zest and the white pith), cut the lime in half lenghtwise (from pole to pole), then cut each part in half lenghtwise, cut off the white pith from the center of the lime, then cut each lime quarter in half crosswise - you should have 8 parts (see the photos in the body of the post).
Place the lime pieces in a small glass, add the sugar and fruits (only if you're making fruit caipirinha). Press the ingredients with a muddler (or alternatively with a wooden spoon) trying to squeze out as much lime juice as possible.
Measure out cachaça and add to the glass. Stir thoroughly with a teaspoon.
Fill the glass with ice cubes or crushed ice and stir once again thoroughly.
Garnish with lime slices (and also fruit pieces if you're making fruit caipirinha). You can also serve the drink with straws. Serve immediately.
Saude! (Cheers!)