Calculus 3 Functions of Several Variables; Limits and Continuity in Higher DimensionsQuestion 2: Read Example 6 (p. 819). Explain how this example shows that it’s not enough to check all the lines going through the origin to show that a limit exists. Include the details involved in this particular example. (a)yk-10k=3k-1➤Xk= -0.1k = -1(b)FIGURE 14.15 (a) The graph off(x, y) = 2x²y/(x² + y²). (b) Along eachpath y = kx² the value of f is constant, butvaries with k (Example 6).Two-Path Test for Nonexistence of a LimitIf a function f(x, y) has different limits along two different paths in the domain off as (x, y) approaches (x, y), then lim(x,y)-(.)f(x,y) does not exist.EXAMPLE 6 Show that the functionf(x, y) =2x²yx++ 32(Figure 14.15) has no limit as (x, y) approaches (0, 0).Solution The limit cannot be found by direct substitution, which gives the indeterminateform 0/0. We examine the values of f along parabolic curves that end at (0, 0). Along thecurve y = kx², x = 0, the function has the constant value2x²(kx²) 2kx*x²4+ (kr²)²x²4+k²x4f(x-3) = -27 -² -² -2x²y=y)+Therefore,2k1+k²12klim f(x, y) - 241₁, [1(x, y) | _] =lim(x, y)-(0,0)(x,y) (0,0)alongy-k1+k²y-kr²This limit varies with the path of approach. If (x, y) approaches (0, 0) along the parabolay = x², for instance, k = 1 and the limit is 1. If (x, y) approaches (0, 0) along the x-axis,k = 0 and the limit is 0. By the two-path test, f has no limit as (x, y) approaches (0, 0).It can be shown that the function in Example 6 has limit 0 along every straight linepath y = mx (Exercise 57). This implies the following observation:Having the same limit along all straight lines approaching (xo.) does not implythat a limit exists at (xo.). Chapter 14 Partial Derivativesf(x, y)At (0, 0), the value of f is defined, but f has no limit as (x, y) → (0, 0). The reason isthat 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" liney = mx, x 0, becauseTy=2xyx² +14.2 Limits and Continuity in Higher Dimensions(x, y)→(0,0)along y-mxTherefore, f has this number as its limit as (x, y) approaches (0, 0) along the line:2m1 + m²lim f(x, y) =2x(mx) 2mx²2m_x² + (mx)²¯¯_x² + m²x² 1 + m²°lim 1) [1(x, 3) ,] =819(x, y)-(0,0)This limit changes with each value of the slope m. There is therefore no single number wemay call the limit of fas (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 morevariables. For a limit to exist at a point, the limit must be the same along every approachpath. This result is analogous to the single-variable case where both the left and right-sidedlimits had to have the same value. For functions of two or more variables, if we ever findpaths with different limits, we know the function has no limit at the point they approach.

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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. a), then lim()-(3) f(x, y) does not exist. EXAMPLE 6 Show that the function f(x, y) = +y² (Figure 14.15) has no limit as (x, y) approaches (0,0). 2r²y

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. a), then lim()-(3) f(x, y) does not exist. EXAMPLE 6 Show that the function f(x, y) = +y² (Figure 14.15) has no limit as (x, y) approaches (0,0). 2r²y

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

Chapter1: Functions

Section1.2: Functions Given By Tables

Problem 32SBE: Does a Limiting Value Occur? A rocket ship is flying away from Earth at a constant velocity, and it...

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