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Understanding and Finding Limits in Calculus: Graphical, Numerical, and Analytical Methods

This lesson focuses on the conditions necessary for a limit to exist and methods for finding limits. Students will learn how to predict limits using tables, charts, and analytical strategies. The lesson emphasizes the local behavior of functions near specific points and introduces four ways to depict limits: graphically, numerically, analytically, and verbally. Emphasis is placed on direct substitution for finding limits, especially in cases where the function may be undefined, such as holes or asymptotes in graphs. Examples and practice problems enhance understanding.

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Understanding and Finding Limits in Calculus: Graphical, Numerical, and Analytical Methods

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  1. Section 5.2.3 Day 2 Limits Review

  2. Lesson Objective: Students will: • Understand the necessary conditions for a limit to exist. • Practice finding limits. • Predict limits from tables.

  3. (1).General Idea: Behavior of a function very near the point where (2). Layman’s Description of Limit(Local Behavior) L If x approaches a, from both sides, then the function approaches a single number, L. a (3). Notation (4). Mantra

  4. 4 Ways to Depict Limits G N A W Graphically Numerically Analytically Words

  5. G N A W Graphically

  6. FINDING LIMITS • Graphically

  7. G N A W • Numerically • Words Mantra:

  8. FINDING LIMITS • Analytically Rem: Always start with Direct Substitution

  9. FINDING LIMITS • Analytically

  10. (1). If a is in the domain: Use Substitution

  11. (2). If a is not in the domain: Start with Substitution A. If the result is , ( this step must be shown), then factor and substitute again.

  12. Graphically Since the function is undefined at x = 3, this produces a “hole” in the graph.

  13. (2). If a is not in the domain: Start with Substitution B. If the result is , ( this step must be shown), then factor and substitute again.

  14. Graphically Since the function is undefined at x = 3, this produces a “hole” and an asymptote in the graph.

  15. Polynomials (Power Functions)

  16. Rational Functions

  17. Rational Functions Divide all terms in the numerator & denominator by the largest degree in the denominator.

  18. Rational Functions Divide all terms in the numerator & denominator by the largest degree in the denominator.

  19. Leading Term Test Take the ratio of the leading terms.

  20. Examples

  21. Examples

  22. Examples

  23. Assignment WS 13.1#7-32

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