Wednesday, October 3, 2018

sequences and series - Using the squeeze theorem liberally to prove a limit



The question is, find the limit of  an=(en+3n)/5n as  n
I tried using L'hopital's rule, but didn't seem to get anything useful, so I figured I may be able to use the squeeze theorem. Would this be an appropriate use of the squeeze theorem, and would there be a better method to proving that the limit approaches zero?



 1/n(en+3n)/5n10/n for all n, with both  1/n and  10/n approaching zero as n approaches infinity.


Answer



Hint: I don't believe the bounds you have are correct. Instead, I suggest something like 0en+3n5n3n+3n5n=23n5n.

These inequalities are much more obvious, hold for all n, thus we can take the limit. What do we get?



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