Infinite number of rooms Hotel Paradox Proofs

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Presentation transcript:

Infinite number of rooms Hotel Paradox Proofs 1/3=.9999999999 2inf = inf implies inf = 0 Intro to Calculus Rainbow Bridge, find the area under the curve Find the instantaneous rate of change or tangent line at an exact point.

AKA, Asymptotes of Rational Functions, Calculus Style 12.4 Limits and Infinity AKA, Asymptotes of Rational Functions, Calculus Style

Part I: Infinite Limits: vertical asymptote at x=0. online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc02_2.ppt

“On the contrary, it tells HOW the limit FAILS to exist.” IMPORTANT NOTE: The equal sign in the statement does NOT mean the limit exists! “On the contrary, it tells HOW the limit FAILS to exist.” Page 80 Dr. Larson & Dr. Hostetler www.vcsc.k12.in.us/lincoln/math/CaseNotes/.../InfiniteLimits.ppt

Definition of a Vertical Asymptote If f(x) approaches infinity or negative infinity as x approaches c from the left or right, then x = c is a vertical asymptote of f. www.vcsc.k12.in.us/lincoln/math/CaseNotes/.../InfiniteLimits.ppt

Digging deeper… Infinity is a very special idea. We know we can't reach it, but we can still try to work out the value of functions that have infinity in them. http://www.mathsisfun.com/calculus/limits-infinity.html

Question: What is the value of 1/∞ ? Answer: We don't know! Maybe we could say that 1/∞ = 0, ... but if we divide 1 into infinite pieces and they end up 0 each, what happened to the 1? In fact 1/∞ is known to be undefined. http://www.mathsisfun.com/calculus/limits-infinity.html

But We Can Approach It! x 1/x 1 1.00000 2 0.50000 4 0.25000 10 0.10000 0.01000 1,000 0.00100 10,000 0.00010 http://www.mathsisfun.com/calculus/limits-infinity.html

The limit of 1/x as x approaches Infinity is 0 Furthermore: http://www.mathsisfun.com/calculus/limits-infinity.html

Limits at Infinity Divide through by the highest power of x in the denominator Simplify Substitute 0 for 1/xn www.mrsantowski.com/MCB4U/Notes/PowerPointNotes/BCC016.ppt

Example Divide by www.rowan.edu/open/depts/math/.../Limits%20and%20Continuity.pp.

More Examples www.rowan.edu/open/depts/math/.../Limits%20and%20Continuity.pp.

www.rowan.edu/open/depts/math/.../Limits%20and%20Continuity.pp.