Yes. is a special limit derived for the purposes of determining the trigonometric derivatives.
Check out the proof here...
World Web Math: Useful Trig Limits
Informally speaking, yes -- when you have a fraction with numerator reaching zero and denominator reaching infinity, the fraction reaches zero. The other statement is true as well : really big number divided by really small number is like really big number times another really big number, equals really really big number!