Question

Can someone explain this to me. In SPivak's Calculus he defines a limit of a function as follows:
"The function f approaches the limit l near a means
for every e > 0 there is some d > 0 such that, for all x, if 0< |x - a|< d, then |f(x) - l| < e."
"When proving that f does not approach l at a, be sure to negate the definition correctly.
He then gives an example "to show sin 1/x does not approach 0 near 0, let e = 1/2 for d>0 some x
obeys 0<|x-a| 13 years ago

Answer 1

thers a typo on last but one line it shld be

'obeys 0<|x - 0| < d ' ( a = 0 , of course)

i just cant get my head around this one - i get the definition ok, but the example is the problem.

Answer 2

limit of sin(1/x) is not defined as x->0, what value of l did you try to use?

Answer 3

yes - he says that it is false that f approaches 0 near 0 - i dont understand the last twp sentences of my post

Answer 4

try putting x=2/pi or x=2n/pi

Answer 5

sorry, x=2/pi(2n+1)

Answer 6

x->0 as n->inf, but sin(1/x) -> 1

Answer 7

right - i see that now - its the way that spivak explains it - why didnt he say sin(1/x) --> 1
thanks

Answer 8

yw ,,, though sin(1/x) -> 1 only as x->2/pi(2n+1)
It's an oscillating sequence
http://www.wolframalpha.com/input/?i=plot+sin%281%2Fx%29+from+-1+to+1

Answer 9

i was lucky to get spivaks book second hand but cant get on with it - can u recommend another book on advanced calculus?

Answer 10

lol ... i've tom apostol advanced calculus .. pdf version

Answer 11

i depend on video lectures

Answer 12

ok thanx