Chapter 2 Limits and Continuity
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.
Properties ofLimits continued Product Rule: Constant Multiple Rule:
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]
Example One-Sided and Two-Sided Limits Find the following limits from the given graph. 4 o 3 1 2
2.2 Limits Involving Infinity
Quick Review Solutions [-12,12] by [-8,8] [-6,6] by [-4,4]
What you’ll learn about • Finite Limits as x→±∞ • Sandwich Theorem Revisited • Infinite Limits as x→a • End Behavior Models • Seeing Limits asx→±∞ …and why Limits can be used to describe the behavior of functions for numbers large in absolute value.
Finite limits as x→±∞ The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, when we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line. When we say “the limit of f as x approaches negative infinity (- ∞)” we mean the limit of f as x moves increasingly far to the left on the number line.
Example Horizontal Asymptote [-6,6] by [-5,5]
Properties of Limits as x→±∞ ProductRule: Constant Multiple Rule:
Example Vertical Asymptote [-6,6] by [-6,6]