From: Stephen Dalton 
Date: December 16, 2009 7:41:32 PM EST
To: James Propp 
Subject: Comments on 92.305 Assignment #10

- With exercise 6.2.5, we are dealing with a sequence of functions the values of which depend explicitly on $n$.  When discussing the limit of the sequence of functions, it does not really work to use conditions like "when $|x| \leq 1/n$" or "when $|x| \geq 1/n," as the limit function is after the $n$'s have all run their course and so it does not make sense anymore to make comparisons between $x$ and expressions involving $n$.  Also, keep in mind that in general, expressions comparing $x$'s to $n$'s may change as $n$ increases (specifically, every $|x|$ will eventually be less than $1/n$ as $n \rightarrow \infty$), and so the condition "when $|x| \geq 1/n$" in particular poses a problem since it will eventually fail for that fixed $x$.  Recall that the limit function is a POINTWISE phenomenon, based on variations in $x$ and $x$ alone; variations in $n$ do not affect the limit function, since the limit function only becomes a consideration once the "$n$-game" has ended.

- In general, students were very good at applying the Weierstrauss M-Test.  It was encouraging to see such fluency in so many papers!

-Steve