I'm asked to find the limit as n approaches infinity of : [(2^(n) +1]/ [2^(n+1)]. The review we were given has [1+ 2^(-n)] / 2 as the next step, but I'm not quite sure how to make that leap myself. The final answer is 1/2. How do you solve this limit?

Question

I'm asked to find the limit as n approaches infinity of : [(2^(n) +1]/  [2^(n+1)]. The review we were given has  [1+ 2^(-n)] / 2 as the next step, but I'm not quite sure how to make that leap myself. The final answer is 1/2. How do you solve this limit?

Steve · Answer

I'd go with this: (2^n+1)/2^(n+1) = 2^n/2^(n+1) + 1/2^(n+1) = 1/2 + 1/2^(n+1) -> 1/2 as n->∞ The suggestion requires you to multiply top and bottom by 2^-n.