CalcNotes02

CalcNotes02 - (Chapter 2: Limits and Continuity) 2.0.1...

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(Chapter 2: Limits and Continuity) 2.0.1 CHAPTER 2: LIMITS AND CONTINUITY In Swokowski (Classic / 5 th ed.) In Thomas (11 th ed.) 2.1: An Introduction to Limits 2.1, 2.3 2.1, 2.4 2.2: Properties of Limits 2.3 2.2 2.3: Limits and Infinity I 2.4 2.4 2.4: Limits and Infinity II 2.4 2.5 2.5: The Indeterminate Forms 0/0 and ± / ± 2.1, 2.3, 2.4 2.1, 2.2, 2.4, 2.5 2.6: The Squeeze (Sandwich) Theorem 2.3 2.2 2.7: Precise Definitions of Limits 2.2 2.3 2.8: Continuity 2.5 2.6 ASSUMPTIONS THROUGHOUT THE NOTES Unless otherwise specified … • We assume that f and g denote functions. • We assume that a , b , c , and k denote real constants. • We assume that the domain of a function is its implied domain. • We assume that graphs extend beyond the scope of the figures in an expected manner, unless endpoints are clearly shown. Arrowheads may help to make this clearer. • Before we get to multivariable calculus, we will assume that “real constants” are “real constant scalars,” as opposed to vectors.
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(Section 2.1: An Introduction to Limits) 2.1.1 SECTION 2.1: AN INTRODUCTION TO LIMITS PART A: “EASY” EXAMPLES Example 1 (Polynomial Function) Assuming fx () = 3 x 2 + x ± 1 , evaluate lim x ± 1 . What is this asking? ± means “approaches.” We will discuss this more rigorously later. lim x ± 1 is read: “the limit of as x approaches 1.” It is the real number that approaches as x approaches 1, if such a number exists. Solution Method f is a polynomial function with presumably unrestricted domain ± . Here, we substitute (“plug in”) x = 1 and evaluate f 1 . Warning 1 : Sometimes, lim x ± a does not equal fa . We will see examples of this later. Solution lim x ± 1 = lim x ± 1 3 x 2 + x ² 1 Warning 2 : When taking the limit of an expression consisting of more than one term, make sure to group the entire expression. = 31 2 + 1 ± 1 Warning 3 : When performing substitutions, be prepared to use grouping symbols unless you are sure that they are not required. = 3
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(Section 2.1: An Introduction to Limits) 2.1.2 Notation We can write: lim x ± 1 fx () = 3 . Alternatively, we can write: ± 3 as x ± 1 . A graph can demonstrate this. Consider the graph of y = ; here, y = 3 x 2 + x ± 1 . (Figure 2.1.a) Imagine that the arrows in the figure above represent two lovers running towards each other along the parabola. What is the y -coordinate of the point that they are approaching? It is 3. (Remember that y -coordinates correspond to function values here.) Example 2 (Rational Function) Assuming = 2 x + 1 x ± 2 , evaluate lim x ± 3 . Solution Method f is a rational function with implied domain x ± ± x ² 2 {} . Here, we observe that 3 is in the domain of f , so we substitute (“plug in”) x = 3 and evaluate f 3 .
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(Section 2.1: An Introduction to Limits) 2.1.3 Solution lim x ± 3 fx () = lim x ± 3 2 x + 1 x ² 2 = 23 + 1 3 ² 2 = 7 A graph can demonstrate this. Consider the graph of y = ; here, y = 2 x + 1 x ± 2 .
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CalcNotes02 - (Chapter 2: Limits and Continuity) 2.0.1...

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