2.1 Rates of Change and Limits. What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided.

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Presentation transcript:

2.1 Rates of Change and Limits

What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwich Theorem …and why Limits can be used to describe continuity, the derivative and the integral: the ideas giving the foundation of calculus.

Average and Instantaneous Speed

Definition of Limit

Definition of Limit continued

Properties of Limits

Properties of Limits continued Product Rule: Constant Multiple Rule:

Properties of Limits continued

Example Properties of Limits

Polynomial and Rational Functions

Example Limits

Evaluating Limits As with polynomials, limits of many familiar functions can be found by substitution at points where they are defined. This includes trigonometric functions, exponential and logarithmic functions, and composites of these functions.

Example Limits

[-6,6] by [-10,10]

One-Sided and Two-Sided Limits

One-Sided and Two-Sided Limits (continued)

Example One-Sided and Two-Sided Limits o Find the following limits from the given graph.

Sandwich Theorem