TheLivingShadow on 28 August 2011
Alright, I'm gonna assume it's the second case. Look:
lim(x->0) (x+tan(x))/sin(x) = lim(x->0) (x/sin(x)) + lim(x->0) (tan(x)/sin(x)). What I did here was separate the fraction sum. Now, recall that tan(x)=sin(x)/cos(x). Then:
= lim(x->0) (x/sin(x)) + lim(x->0) [(sin(x)/cos(x))/sin(x)]. The bold term is 1 (from the first limit identity you wrote) and the italics cancel out. Then the expression turns into:
= 1 + lim(x->0)(1/cos(x)). But note that as x goes to 0, cos(x) goes to 1. So:
= 1 + 1/1 = 2
Your answer is 2.
