Spivak proves that limit of function f (x) as x approaches a is always unique. ie...If lim f (x) =l x-> a and lim f (x) =m x-> a Then l=m. This definition means that limit of function can't approach two different values. He takes definition of both the limits. He says for first limit, we can find some δ1> 0 for every ε> 0 Such that if 0 <|x-a|<δ1, then |f (x)-l|<ε and also some δ2> 0 for every ε> 0 Such that 0 <|x-a|<δ2, then |f (x)-m|<ε He then proves by saying a delta which works for one definition may not work for another and since both in qualities must be satisfied, he takes δ=min (δ1, δ2) My questions here are as follows: 1. Why did he take only one ε in both definitions but 2 δs in both defintions? Shouldn't there be ε1 and ε2? 2. He then says since 0 <|x-a|<δ1 and 0 <|x-a|<δ2 are both true, so 0 <|x-a|<min (δ1, δ2) By definition. My question is what if δ1 is exclusively satisfied by first sentence alone and not second sentence, though δ1 <δ2? Also what if δ2 is satisfied by second sentence alone and not the first one, even if δ2 <δ1. How can we guarantee that atlest one of δ1 or δ2 satisfy both the statements, leave alone the prospect of taking minimum? To be clear, say, if δ2>δ1, so δ=δ1 which is the minimum value. Then how can we prove that such a δ also satisfies second statement? Sorry, if my questions sounds silly.