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Limits Involving Infinity in Calculus: Exploring Horizontal Asymptotes & End Behavior Models

This article explores limits involving infinity in calculus, specifically focusing on finding horizontal asymptotes and end behavior models. It covers the definition of a horizontal asymptote, the three possibilities for rational functions, and how to find limits using the Sandwich Theorem. The properties of limits are also discussed, along with examples and the use of substitution.

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Limits Involving Infinity in Calculus: Exploring Horizontal Asymptotes & End Behavior Models

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Presentation Transcript

  1. Warm-up • Evaluate:

  2. 2.2 Limits involving Infinity Calculus

  3. Limits as • “The limit as x approaches infinity” means the limit of f as x moves increasingly far to the right on a number line.

  4. Definition of a Horizontal Asymptote

  5. Finding horizontal asymptotes • Use graphs to find , and identify all horizontal asymptotes of . • so there are horizontal asymptotes at

  6. End behavior model • For Rational Functions we have three possibilities: • degree on top = degree on bottom • degree on top < degree on bottom (bottom heavy) • degree on top > degree on bottom (top heavy)

  7. Finding a limit as x approaches • We will use the Sandwich Theorem to find . • Since • , so • You can and should memorize this.

  8. Properties of Limits

  9. Example • Find

  10. End behavior model • Which part of the function determines what happens at • An End Behavior Model (EBM) tells us what the function is doing as x approaches ± • In a polynomial, it’s the term with the largest exponent. • looks the same as for large values of x so “” is the end behavior model.

  11. Example • Find the end behavior model(s) of the following function. Then find the limits as .

  12. Using substitution • Find. • Look at the graph. What do you think the limit is? • Big idea:

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