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.

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Presentation transcript:

Infinite number of rooms Hotel Paradox Proofs 1/3= inf = 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

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

IMPORTANT NOTE: The statement does NOT mean the limit exists! “On the contrary, it tells HOW the limit FAILS to exist.”

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.

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.

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.

But We Can Approach It! x1/x , ,

The limit of 1/x as x approaches Infinity is 0 Furthermore:

Limits at Infinity Divide through by the highest power of x in the denominator Simplify Substitute 0 for 1/x n 1.

Example Divide by 2.

More Examples 3.

or DNE 4. 5.

Asymptotes of a Rational Functions (Limits at Infinity) Let f be the rational function, N(x) and D(x) have no common factors. Vertical Asymptotes occur at the zeros of D(x), V.A. : Set D(x) =0 Horizontal Asymptote 1. If, the graph approaches the x-axis. H.A. : y=0 2. If n = d the graph has a horizontal asymptote of H.A. : y = the ratio of the leading coefficients. 3. If, the graph has no horizontal asymptote.

Vertical Asymptotes occur at the zeros of D(x), V.A. : Set D(x) =0 Horizontal Asymptote 1. If the graph approaches the x-axis. H.A. : y=0 2. If n = d the graph has a horizontal asymptote of H.A. : y = the ratio of the leading coefficients. 3. If, the graph has no horizontal asymptote. Oh, so limits is how we got our rules for horizontal asymptotes !

Limits at Infinity 1 0 or DNE

Find the limit ASYMPTOTE STYLE! =1 =-3/4 =2 =0 DNE

12-4 HW

Find the limit (if it exists). =0 =2