lime (n.1) Look up lime at
"chalky, sticky mineral used in making mortar," from Old English lim "sticky substance, birdlime;" also "mortar, cement, gluten," from Proto-Germanic *leimaz (source also of Old Saxon, Old Norse, Danish lim, Dutch lijm, German Leim "birdlime"), from PIE root *(s)lei- "slime, slimy, sticky" (source also of Latin limus "slime, mud, mire," linere "to smear;" see slime (n.)).

Bird-lime is prepared from the bark of the holly, it was spread on twigs and used for catching small birds. The lime used in building, etc. is made by putting limestone or shells in a red heat, which burns off the carbonic acid and leaves a brittle white solid which dissolves easily in water. Hence lime-kiln (late 13c.), lime-burner (early 14c.). As a verb, c. 1200, from the noun.
lime (n.3) Look up lime at
"linden tree," 1620s, earlier line (c. 1500), from Middle English lynde (early 14c.), from Old English lind "lime tree" (see linden). Klein suggests the change of -n- to -m- began in compounds whose second element began in a labial (such as line-bark, line-bast). An ornamental European tree, it is unrelated to the tree that produces the citrus fruit.
lime (n.2) Look up lime at
greenish-yellow citrus fruit, 1630s, probably via Spanish lima, from Arabic limah "citrus fruit," from Persian limun "lemon" (see lemon (n.1)). Lime-green as a color is from 1890.