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What is the used of limit? Why do we need to find the limit of a function? Why is it that the limit is 4?

What is the used of limit? Why do we need to find the limit of a function? Why is it that the limit is 4?

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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What is the used of limit? Why do we need to find the limit of a function? Why is it that the limit is 4?

Definition of Limit We begin by investigating the behavior of the function f defined by f(x) = x² = x + 2 for values of x near 2. The following tables give values of f(x) for values of x close to 2 but not equal to 2. f(x) 1.0 2.000000 1.5 2.750000 1.8 3.440000 1.9 3.710000 1.95 3.852500 1.99 3.970100 1.995 3.985025 1.999 3.997001 x f(x) 3.0 8.000000 2.5 5.750000 2.2 4.640000 2.1 4.310000 2.05 4.152500 2.01 4.030100 2.005 4.015025 2.001 4.003001 X f(x) approaches 4... FIGURE 1 0 lim (x²-x + 2) = 4 x-2 y = x² = x + 2 -2- ... as x approaches 2 X From the table and the graph of f (a parabola) shown in Figure 1 we see that when x is close to 2 (on either side of 2), f(x) is close to 4. In fact, it appears that we can make the values of f(x) as close as we like to 4 by taking x sufficiently close to 2. We express this by saying "the limit of the function f(x) = x² − x + 2 as x approaches 2 is equal to 4." The notation for this is -
Transcribed Image Text:Definition of Limit We begin by investigating the behavior of the function f defined by f(x) = x² = x + 2 for values of x near 2. The following tables give values of f(x) for values of x close to 2 but not equal to 2. f(x) 1.0 2.000000 1.5 2.750000 1.8 3.440000 1.9 3.710000 1.95 3.852500 1.99 3.970100 1.995 3.985025 1.999 3.997001 x f(x) 3.0 8.000000 2.5 5.750000 2.2 4.640000 2.1 4.310000 2.05 4.152500 2.01 4.030100 2.005 4.015025 2.001 4.003001 X f(x) approaches 4... FIGURE 1 0 lim (x²-x + 2) = 4 x-2 y = x² = x + 2 -2- ... as x approaches 2 X From the table and the graph of f (a parabola) shown in Figure 1 we see that when x is close to 2 (on either side of 2), f(x) is close to 4. In fact, it appears that we can make the values of f(x) as close as we like to 4 by taking x sufficiently close to 2. We express this by saying "the limit of the function f(x) = x² − x + 2 as x approaches 2 is equal to 4." The notation for this is -
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