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lim n to infinity (1/1 + 5n)

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2 Answers

                  1           
   lim    ( ———— ) = 
n → ∞     1 + 5n       

( 1 + 5n ≠ 0 ---> n ≠ - 1/5 )  

           lim 1
= ———————— =   
     lim 1 + lim 5n

             1
= ———————— = 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