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.)
Daniel J. Teague Department of Mathematics NC School of Science and Mathematics 1219 Broad Street Durham, NC 27705 email@example.com
-----Original Message----- From: Cindy Couchman [mailto:firstname.lastname@example.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