I was just wondering while in the shower if the following limit

Can be done another way...Please...Please don't put down the ways that are commonly seen...I know that

increases much faster than thus the limit is 0...but does this method work?

For aesthetics n will denote ∞

So rewriting

as

Factoring out an from top and bottom we get

Cancelling we would get

Which would be equilvalent to

Is that mathematically sound?