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10 Introduction to the Derivative Copyright © Cengage Learning. All rights reserved. 10.3 Limits and Continuity: Algebraic Approach Copyright © Cengage Learning. All rights reserved. Limits and Continuity: Algebraic Approach Closed-Form Functions A function is written in closed form if it is specified by combining constants, powers of x, exponential functions, radicals, logarithms, absolute values, trigonometric functions (and some other functions we do not encounter in this text) into a single mathematical formula by means of the usual arithmetic operations and composition of functions. A closed-form function is any function that can be written in closed form. 3 Limits and Continuity: Algebraic Approach Quick Examples 1. 3x2 – |x| + 1, and are written in closed form, so they are all closed-form functions. –1 if x ≤ –1 2. f(x) = x2 + x if –1 < x ≤ 1 2 – x if 1 < x ≤ 2 is not written in closed-form because f (x) is not expressed by a single mathematical formula. 4 Limits and Continuity: Algebraic Approach Theorem 3.1 Continuity of Closed-Form Functions Every closed-form function is continuous on its domain. Thus, if f is a closed-form function and f(a) is defined, we have limx→a f(x) = f(a). Quick Example f(x) = 1/x is a closed-form function, and its natural domain consists of all real numbers except 0. Thus, f is continuous at every nonzero real number. That is, provided a ≠ 0. 5 Example 1 – Limit of a Closed-Form Function at a Point in Its Domain Evaluate algebraically. Solution: First, notice that (x3 – 8)/(x – 2) is a closed-form function because it is specified by a single algebraic formula. Also, x = 1 is in the domain of this function. Therefore, 6 Limits and Continuity: Algebraic Approach Functions with Equal Limits If f(x) = g(x) for all x except possibly x = a, then Quick Example for all x except x = 1. Therefore, 7 Limits at Infinity 8 Example 5 – Limits at Infinity Compute the following limits, if they exist: 9 Example 5 – Solution Solution for a and b. The highest power of x in both the numerator and denominator dominate the calculations. For instance, when x = 100,000, the term 2x2 in the numerator has the value of 20,000,000,000, whereas the term 4x has the comparatively insignificant value of 400,000. Similarly, the term x2 in the denominator overwhelms the term –1. 10 Example 5 – Solution cont’d In other words, for large values of x (or negative values with large magnitude), Use only the highest powers top and bottom. Therefore, 11 Example 5 – Solution cont’d c. Applying the previous technique of looking only at highest powers gives Use only the highest powers top and bottom. Simplify. As x gets large, –x/2 gets large in magnitude but negative, so the limit is 12 Example 5 – Solution d. cont’d Use only the highest powers top and bottom. As x gets large, 2/(5x) gets close to zero, so the limit is 13 Example 5 – Solution cont’d e. Here we do not have a ratio of polynomials. However, we know that, as t becomes large and positive, so does e0.1t, and hence also e0.1t – 20. Thus, 14 Example 5 – Solution f. As t → , the term (3.68)–t = cont’d in the denominator, being 1 divided by a very large number, approaches zero. Hence the denominator 1 + 2.2(3.68)–t approaches 1 + 2.2(0) = 1 as t → . Thus, 15 Limits at Infinity Theorem 3.2 Evaluating the Limit of a Rational Function at If f(x) has the form with the ci and di constants (cn 0 and dm 0), then we can calculate the limit of f(x) as x → by ignoring all powers of x except the highest in both the numerator and denominator. Thus, 16 Limits at Infinity Quick Example 17 Some Determinate and Indeterminate Forms 18 Some Determinate and Indeterminate Forms Some Determinate and Indeterminate Forms are indeterminate; evaluating limits in which these arise requires simplification or further analysis. The following are determinate forms for any nonzero number k: 19 Some Determinate and Indeterminate Forms and, if k is positive, then 20 Some Determinate and Indeterminate Forms Quick Examples 1. 2. 3. 21