Find the limit using limit theorem
lim n to infinity (1/1 + 5n)
lim ( ———— ) =
n → ∞ 1 + 5n
( 1 + 5n ≠ 0 ---> n ≠ - 1/5 )
= ———————— =
lim 1 + lim 5n
= ———————— = 0
1 + 5 lim (n)
As n approaches infinity, multiplying it by 5 doesn't do anything, and adding 1 doesn't do anything (both because you can't get anything larger than the concept of infinity), and the expression behaves like 1/n.
As n approaches infinity, 1/n approaches 0 (0 might be interpreted as an infinitely small number).
I've never heard the above property of 1/n described as a theorem, though it is well established.
and this (http://www1.maths.leeds.ac.uk/~kisilv/courses/math150.html) website does describe it as a theorem.
If your numerator had an n term in it, then the "5*n" might be significant because the rate at which n approaches infinity (5 times as fast as just "n") might come into play.
Ex. lim n -> infinity (2n/5n+1) = 2/5