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#1 2007-06-29 00:00:41

uinaik
Member
Registered: 2007-06-28
Posts: 0

Solve the following

Examine the continuity at 0 of f such that f(x)=sin (1/x) when x is not 0 and f(0)=0

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#2 2007-06-29 01:11:43

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Solve the following

There are a couple of different ways to do this.  The first is through rigorous analysis.  I get the feeling you aren't in an analysis course, so I'm going to skip that.

The second, is to first note that:

lim x->infinity sin(x)

Does not exist.  Sin(x) doesn't converge to any single point, rather it's always spread out through the interval [-1, 1].  So we can say:

lim x->0 sin(1/x)

Also does not exist.  Why can we say this?  Simply becaue:

lim x->infinity 1/x = 0

So we've established that this limit does not exist.  But remember what the requirement for continuity is:

lim x->c f(x) = f(c)

We want to see if:

lim x->0 sin(1/x) = f(0) = 0

Is it possible for "does not exist" to equal 0?  Of course not.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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