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Topic: [ap-calculus] limit question
Replies: 1   Last Post: Aug 23, 2012 9:38 AM

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Teague, Dan

Posts: 2,314
Registered: 12/6/04
RE: [ap-calculus] limit question
Posted: Aug 23, 2012 9:38 AM
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Cindy,

Another example that makes for an interesting calculator investigation is to consider the limit as x -> 0 of (1-cos(x^2))/(x^4).

Graph y = (1-cos(x^2))/(x^4) on x in [-5, 5], y in [-1, 1].
Now change the viewed domain to [-1, 1], then [-.1, .1], [-.01,.01], and [-.001, .001].

You will find that the calculator will evaluate cos(x^2) as 1 long before it evaluates x as 0. (This example comes from Smith & Minton's Calculus text.)

Dan

Daniel J. Teague
Department of Mathematics
NC School of Science and Mathematics
1219 Broad Street
Durham, NC 27705
teague@ncssm.edu


-----Original Message-----
From: Cindy Couchman [mailto:ccouchman@buhlerschools.org]
Sent: Wednesday, August 22, 2012 1:02 PM
To: AP Calculus
Subject: [ap-calculus] limit question

Greetings!
We are investigating graphically and numerically the following limit today:

Limit as x approaches 0 of
(1 + x)^(1/x)

The limit by table goes to 2.7183 (e). However, when a student evaluated 10^-50 power, the calculator said it was 1 - which makes sense as you will get 1 raised to a very large number. So, what exactly IS the limit of this equation? Graphically, it appears there is a hole in the graph and not a value of 1 graphed at 0. I am puzzled on what is happening here. HELP?! Thank you!
Cindy Couchman

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