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Topic: [ap-calculus] Limit Question - Sandwhich Theorem
Replies: 2   Last Post: Aug 24, 2012 4:44 PM

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Dave L. Renfro

Posts: 2,165
Registered: 11/18/05
re:[ap-calculus] Limit Question - Sandwhich Theorem
Posted: Aug 24, 2012 4:44 PM
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Jean Contreras wrote (in part):

> Also is the Sandwich theorem only really used with sine
> and cosine? If not does anyone have some other good examples.

Let f be the function defined by f(x) = 0 if x is rational
and f(x) = 1 if x is irrational. Note that f is discontinuous
at every point. In fact, at every point f has neither a left
limit nor a right limit.

Now define the function g by putting g(x) = x*f(x).

Using the Sandwich Theorem, you can show that g is
continuous at x = 0. It's also easy to show that g
is not continuous at every other point, so g has the
property that it's continuous at exactly one point.

Dave L. Renfro
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