Prove the following limit. lim x → 5 3x − 8 = 7 solution 1. preliminary analysis of the problem (guessing a value for δ). let ε be a given positive number. we want to find a number δ such that if 0 < |x − 5| < δ then |(3x − 8) − 7| < ε. but |(3x − 8) − 7| = |3x − 15| = 3 . therefore, we want δ such that if 0 < |x − 5| < δ then 3 < ε that is, if 0 < |x − 5| < δ then < ε 3 . this suggests that we should choose δ = ε/3. 2. proof (showing that δ works). given ε > 0, choose δ = ε/3. if 0 < < δ, then |(3x − 8) − 7| = = 3 < 3δ = 3 = ε. thus if 0 < |x − 5| < δ then |(3x − 8) − 7| < ε. therefore, by the definition of a limit lim x → 5 3x − 8 = 7.