Prove that lim(n->infinity) (1/n) = 0. Make a proof with definition of limit.

Question

Make a proof about definition of limit.

Answer 1

I gather you are looking for the epsilon def'n of the limit...

 

For any ε > 0 you need to find an integer N > 0 so that if n > N then | 1 / n - 0 |< ε---> 1 / ε < n

...so choose N > 1/ε

Answer 2

1/n, break the numerator and denominator into separate parts.  As n—>infinity, the numerator remains unchanged.  So we need only to worry about the denominator.   The denominator approaches infinity.  1/∞ = 0.

You may also consider L'Hopital's rule.  Take the derivative of the denominator, 1 (a constant) and the derivative is 0.  Take the derivative of the denominator, n, and it becomes 1.  The derivative of the numerator divided by the derivative of the denominator  is 0.