By definition:
a is the limit of a function f(x) when x goes to infinity if ∀ε>0 ∃c: ∀x>c |f(x)-a|<ε.
Then I will claim that there exists c=1/ε, such that for any x>c |1/x-0|=1/|x|<1/c=1/(1/ε)=ε.
I just showed that no matter what ε is, I can always pick up a point c, in this case c=1/ε, such that the value of a function differs less than epsilon from zero for any x>c. This means that the limit of a function is zero as x goes to infinity.
This and the second problem you posted are from Calculus I.