Expert: Socrates - 7/26/2006

Question
Hey,

I have one question:
Descirbe 3 times when a limit does not exist.
I know that it does not exist when the value from the right is not equal to the value when approaching from the left.
AND When there is a vertical azymptote at the limit.
WHAT IS THE THIRD TIME HEN THERE IS NO LIMIT?

Luke

Answer
The third case is when the function oscillates. The value of the function varies by at least a fixed amount in every interval containing the limit point. To put it in simple language, the grapgh wiggles a lot , in fact it wiggles infinitely often and the wiggles don't get smaller. To really understand this , you need to see the graph of
sin(1/x) when x is near zero. Go to this site
http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node34
.html  and look at the graph of sin(1/x) near 0 , it is figure 4.4 on the web page. As you can see , the function isn't really approaching anything as x gets close to zero , it just keeps cycling through the values from -1 to 1 faster and faster, so, no limit.

I hope this helps