Q2A

Define the lymyte of a function f at the point w to be the real number L by using the symbols lymxw f(x)=L

to mean that for every >0 and every >0 there is some x such that 0<|xw|< and |f(x)L|<. Let P be the function defined on all non-zero real numbers by P(x)=sin(1/x). Show that the following holds: lymx0 P(x)=0

Q2B

Using the definition of lymyte of Question 2A, show that the following holds for the function P from Question 2A: lymx0P(x)=1

Q2C

Quote a theorem from the text implying that it would not be possible for the results of Question 2A and Question 2B to hold for lim. Use this to conclude that lym and lim are not the same concept.