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2xy 14.2 Limits and Continuity in Higher Dimensions At (0, 0), the value of f is defined, but f has no limit as (x, y) → (0, 0). The reason is that different paths of approach to the origin can lead to different results, as we now see. For every value of m, the function f has a constant value on the "punctured" line y=mx, x 0, because f(x, y) +y 2x(mx) 2mx² 2m *x² + (mx)² ¯ x² + m²x² 1 + m²° Therefore, f has this number as its limit as (x, y) approaches (0,0) along the line: [ f(x, y) |___ ] - 2m 1+ m² 819 lim f(x, y) = lim (z. y) +(0,0) (x, y)-(0,0) alongy-m This limit changes with each value of the slope m. There is therefore no single number we may call the limit off as (x, y) approaches the origin. The limit fails to exist, and the func- tion is not continuous at the origin. Examples 4 and 5 illustrate an important point about limits of functions of two or more variables. For a limit to exist at a point, the limit must be the same along every approach path. This result is analogous to the single-variable case where both the left- and right-sided limits had to have the same value. For functions of two or more variables, if we ever find paths with different limits, we know the function has no limit at the point they approach. Two-Path Test for Nonexistence of a Limit If a function f(x, y) has different limits along two different paths in the domain of f as (x, y) approaches (to-yo), then lim()-() f(x, y) does not exist.

2xy 14.2 Limits and Continuity in Higher Dimensions At (0, 0), the value of f is defined, but f has no limit as (x, y) → (0, 0). The reason is that different paths of approach to the origin can lead to different results, as we now see. For every value of m, the function f has a constant value on the "punctured" line y=mx, x 0, because f(x, y) +y 2x(mx) 2mx² 2m *x² + (mx)² ¯ x² + m²x² 1 + m²° Therefore, f has this number as its limit as (x, y) approaches (0,0) along the line: [ f(x, y) |___ ] - 2m 1+ m² 819 lim f(x, y) = lim (z. y) +(0,0) (x, y)-(0,0) alongy-m This limit changes with each value of the slope m. There is therefore no single number we may call the limit off as (x, y) approaches the origin. The limit fails to exist, and the func- tion is not continuous at the origin. Examples 4 and 5 illustrate an important point about limits of functions of two or more variables. For a limit to exist at a point, the limit must be the same along every approach path. This result is analogous to the single-variable case where both the left- and right-sided limits had to have the same value. For functions of two or more variables, if we ever find paths with different limits, we know the function has no limit at the point they approach. Two-Path Test for Nonexistence of a Limit If a function f(x, y) has different limits along two different paths in the domain of f as (x, y) approaches (to-yo), then lim()-() f(x, y) does not exist.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Calculus 3 Functions of Several Variables; Limits and Continuity in Higher Dimensions

Question 3: Read Example 5 and the boxed text “Two-Path Test for Nonexistence of a Limit” (p. 
818 – 819). Explain what the two-path test says and how this shows that the limit in this example 
does not exist at the origin. Include the details involved in this particular example.

14.2 Limits and Continuity in Higher Dimensions At (0, 0), the value of f is defined, but f has no limit as (x,y) → (0, 0). The reason is that different paths of approach to the origin can lead to different results, as we now see. For every value of m, the function f has a constant value on the "punctured" line y=mx, x 0, because 5(x, y) = 2 + 1 ² |-- 2xy f(x,y) 2x(mx) 2mx² 2m x² + (mx)² x² + m²x² 1 + m² 819 Therefore, f has this number as its limit as (x, y) approaches (0, 0) along the line: 2m lim f(x, y) = lim [1(x, y) = 1/² (x, y)→(0,0) 1 + m² (x, y)→(0,0) along m This limit changes with each value of the slope m. There is therefore no single number we may call the limit off as (x, y) approaches the origin. The limit fails to exist, and the func- tion is not continuous at the origin. Examples 4 and 5 illustrate an important point about limits of functions of two or more variables. For a limit to exist at a point, the limit must be the same along every approach path. This result is analogous to the single-variable case where both the left- and right-sided limits had to have the same value. For functions of two or more variables, if we ever find paths with different limits, we know the function has no limit at the point they approach. Two-Path Test for Nonexistence of a Limit If a function f(x, y) has different limits along two different paths in the domain of f as (x, y) approaches (xo, ya), then lim(x, y)-(,) f(x, y) does not exist.
Transcribed Image Text:14.2 Limits and Continuity in Higher Dimensions At (0, 0), the value of f is defined, but f has no limit as (x,y) → (0, 0). The reason is that different paths of approach to the origin can lead to different results, as we now see. For every value of m, the function f has a constant value on the "punctured" line y=mx, x 0, because 5(x, y) = 2 + 1 ² |-- 2xy f(x,y) 2x(mx) 2mx² 2m x² + (mx)² x² + m²x² 1 + m² 819 Therefore, f has this number as its limit as (x, y) approaches (0, 0) along the line: 2m lim f(x, y) = lim [1(x, y) = 1/² (x, y)→(0,0) 1 + m² (x, y)→(0,0) along m This limit changes with each value of the slope m. There is therefore no single number we may call the limit off as (x, y) approaches the origin. The limit fails to exist, and the func- tion is not continuous at the origin. Examples 4 and 5 illustrate an important point about limits of functions of two or more variables. For a limit to exist at a point, the limit must be the same along every approach path. This result is analogous to the single-variable case where both the left- and right-sided limits had to have the same value. For functions of two or more variables, if we ever find paths with different limits, we know the function has no limit at the point they approach. Two-Path Test for Nonexistence of a Limit If a function f(x, y) has different limits along two different paths in the domain of f as (x, y) approaches (xo, ya), then lim(x, y)-(,) f(x, y) does not exist.
-0.8 0.8 -0.8 0.8 0 (b) f(x, y) = 0.8 -0.8 -0.8 0 -0.8 -1 FIGURE 14.14 (a) The graph of 2xy x² + 0, (x, y) = (0,0) (x, y) = (0,0). The function is continuous at every point except the origin. (b) The values of f are different constants along each line y = mx, x = 0) (Example 5). DEFINITION A function f(x, y) is continuous at the point (xo, yo) if 1. f is defined at (x, y), 2. lim f(x, y) exists, f(x, y) = f(xo. Jo). lim (x,y)-(3) A function is continuous if it is continuous at every point of its domain. 3. As with the definition of limit, the definition of continuity applies at boundary points as well as interior points of the domain of f. The only requirement is that each point (x, y) near (x, y) be in the domain of f. A consequence of Theorem I is that algebraic combinations of continuous functions are continuous at every point at which all the functions involved are defined. This means that sums, differences, constant multiples, products, quotients, and powers of continuous functions are continuous where defined. In particular, polynomials and rational functions of two variables are continuous at every point at which they are defined. EXAMPLE 5 Show that 2xy (x, y) (0,0) (x, y) = (0,0) is continuous at every point except the origin (Figure 14.14). Solution The function f is continuous at every point (x, y) except (0, 0) because its val- ues at points other than (0, 0) are given by a rational function of x and y, and therefore at those points the limiting value is simply obtained by substituting the values of x and y into that rational expression. f(x, y) = x² + 0,
Transcribed Image Text:-0.8 0.8 -0.8 0.8 0 (b) f(x, y) = 0.8 -0.8 -0.8 0 -0.8 -1 FIGURE 14.14 (a) The graph of 2xy x² + 0, (x, y) = (0,0) (x, y) = (0,0). The function is continuous at every point except the origin. (b) The values of f are different constants along each line y = mx, x = 0) (Example 5). DEFINITION A function f(x, y) is continuous at the point (xo, yo) if 1. f is defined at (x, y), 2. lim f(x, y) exists, f(x, y) = f(xo. Jo). lim (x,y)-(3) A function is continuous if it is continuous at every point of its domain. 3. As with the definition of limit, the definition of continuity applies at boundary points as well as interior points of the domain of f. The only requirement is that each point (x, y) near (x, y) be in the domain of f. A consequence of Theorem I is that algebraic combinations of continuous functions are continuous at every point at which all the functions involved are defined. This means that sums, differences, constant multiples, products, quotients, and powers of continuous functions are continuous where defined. In particular, polynomials and rational functions of two variables are continuous at every point at which they are defined. EXAMPLE 5 Show that 2xy (x, y) (0,0) (x, y) = (0,0) is continuous at every point except the origin (Figure 14.14). Solution The function f is continuous at every point (x, y) except (0, 0) because its val- ues at points other than (0, 0) are given by a rational function of x and y, and therefore at those points the limiting value is simply obtained by substituting the values of x and y into that rational expression. f(x, y) = x² + 0,
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