Chapter 3: Functions of Several Variables
Section 3.2: Limits and Continuity
Essentials
Limit
Definition of the Bivariate Limit
It is first necessary to distinguish between interior and boundary points of the domain of a function f.
A point P is an interior point of the domain of a function f if P is contained in a neighborhood that lies completely in the domain of f.
A point P is a boundary point of the domain of a function f if every neighborhood of P contains points that are in the domain and points that are not in the domain.
Definition 3.2.1 formalizes the meaning of a limit in the plane, called in this guide, the bivariate limit.
Definition 3.2.1: The Bivariate Limit
If P:a,b is an interior point of the domain of f, then the number L is the bivariate limit of fx,y at P, that is, limx,y→P fx,y=L, when, for every number ε>0 there is a corresponding number δ with the property that
0<x−a2+y−b2<δ ⇒fx,y−L<ε
If P:a,b is a boundary point of the domain of f, then limx,y→P fx,y=L, when, for every number ε>0 there is a corresponding number δ with the property that
0<x−a2+y−b2<δ and x,y is in the domain of f ⇒fx,y−L<ε
The points satisfying 0<x−a2+y−b2<δ are said to lie in a deleted neighborhood of a,b. This deleted neighborhood is actually the interior of an annulus that is an open disk of radius δ with the center point a,b removed.
Conceptually, Definition 3.2.1 is the generalization of the formal definition of a limit along the real line: the values of f can be made arbitrarily close to L by the expedient of taking x,y sufficiently close to P. Thus, L is the limit of fx,y at P if all the points in the deleted neighborhood of P produce function values that are close to L.
The distinction between interior and boundary points amounts to this: The limit is taken over the domain of the function.
Proving that L Is the Limit
It is generally difficult to prove that L is the limiting value of fx,y because the requisite estimates demand a facility with manipulating inequalities.
Table 3.2.1 lists several inequalities that are useful for proving that a real number L is indeed the bivariate limit of fx,y. Inequality 3 is the "triangle" inequality. Inequalities 4, 5, 6, and 7 should be self-evident. A proof of Inequality 2 uses Inequality 1, which itself is proved in Example 3.2.28.
Reference
Inequality
Inequality 1
x y ≤x2+y2/2
Inequality 2
|x|+y ≤2 x2+y2
Inequality 3
|x+y|≤x+y
Inequality 4
x≤x2+y2
Inequality 5
y≤x2+y2
Inequality 6
x2≤x2+y2
Inequality 7
y2≤x2+y2
Table 3.2.1 Useful inequalities
The Bivariate Limit in Maple
Maple's limit command can determine the bivariate limit of rational functions in two variables. Access to this functionality is provided through the Context Panel in the option Limit (Bivariate).
Showing the Limit Does Not Exist
It is far easier to show that fx,y does not have a limit at a,b. Recall that for a limit on the line to exist, both left-hand and right-hand limits must exist, and be equal. The same idea holds for the bivariate limit. If the limits taken along two different paths are not equal, then the (bivariate) limit cannot exist. Consequently, to show a bivariate limit does not exist at a,b, it suffices to show that along two different paths through a,b the limits differ, or that one such limit does not exist.
Typical trial paths are the axes, lines y=m x, parabolas y=x2 and x=y2, and on rare occasions, the curves y=x3/2 and y=x2/3.
Continuity
Just as with functions of a single variable, continuity is defined in terms of the limit. Essentially, a function is continuous at a point if it is defined at that point, and its defined value equals its bivariate limit at that point. If the limit point is interior to the domain of the function, the first part of Definition 3.2.1 applies; if a boundary point, the second.
Definition 3.2.2: Continuity at a Point
The function f is continuous at P:a,b if limx,y→P f =fa,b.
If f is continuous at every point in its domain, then f is said to be a continuous function on that domain.
Just as with functions of a single variable, composition of continuous functions results in a continuous function. This is formalized in Theorem 3.2.1.
Theorem 3.2.1: Composition of Continuous Functions
fx,y is continuous at P:a,b
gx is continuous at fa,b
h=g∘f so that hx,y=gfx,y
⇒
h is continuous at P
Examples
Let P be the generic point x,y and O, the origin, 0,0.
For each f in Examples 3.2.(1-10), show that limP→Ofx,y, the bivariate limit at the origin, does not exist.
Example 3.2.1
f=x2x2+y2
Example 3.2.6
f=x−yx+y
Example 3.2.2
f=x2−y2x2+y2
Example 3.2.7
f=x yx4+y4
Example 3.2.3
f=x−yx2+y2
Example 3.2.8
f=x3y2x6+y4
Example 3.2.4
f=x y2x2+y4
Example 3.2.9
f=x4+2 x2y2+3 x y3x2+y22
Example 3.2.5
f=x2+yx2+y2
Example 3.2.10
f=x4y4x2+y43
Example 3.2.11
Show that for f=x yx2+y2 the bivariate limit at the origin does not exist, but the iterated limits limy→0limx→0f and limx→0limy→0f are both zero.
Example 3.2.12
Show that for f=x yx3+y2 the bivariate limit at the origin does not exist, but the iterated limits limy→0limx→0f and limx→0limy→0f are both zero.
Example 3.2.13
Prove that the bivariate limit at the origin for f=x3x2+y2 is zero.
Example 3.2.14
Prove that the bivariate limit at the origin for f=2 x3−y3x2+y2 is zero.
Example 3.2.15
Prove that the bivariate limit at the origin for f=x y x2−y2x2+y2 is zero.
Example 3.2.16
Prove that the bivariate limit at the origin for f=x2y2x2+y2 is zero.
Example 3.2.17
Prove that the bivariate limit at the origin for f=x4+y4x2+y2 is zero.
Example 3.2.18
Prove that the bivariate limit at the origin for f=2 x5+2 y32 x2−y2x2+y22 is zero.
Example 3.2.19
If fx,y=2 x2−6 x y+5 y2, prove that x2+y2<ε/8=δ ⇒ |fx,y|<ε.
Example 3.2.20
Prove the inequality x3−y3≤x2+y23/2.
Example 3.2.21
Show that for f=x sin1/y+y sin1/x the bivariate limit at the origin is zero, but both of the iterated limits limy→0limx→0f and limx→0limy→0f fail to exist. Hint: Show f≤x+y≤2x2+y2.
Example 3.2.22
Show that for f=x yx2+y2+x sin1/y the bivariate limit at the origin and the iterated limit limx→0limy→0f both fail to exist, but the iterated limit limy→0limx→0f is zero.
Example 3.2.23
Show that the bivariate limit at the origin for f={xy2e−|x|/y2y≠00y=0 does not exist.
Example 3.2.24
Extend f=x2⁢y⁢cos⁡x⁢yx2+y2 to a function gx,y that is continuous at the origin.
Example 3.2.25
Extend f=tan⁡x⁢ytan⁡x⁢tan⁡y to a function gx,y that is continuous at the origin.
Example 3.2.26
Extend f=x⁢sin⁡x⁢y2−cos⁡x−cos⁡y to a function gx,y that is continuous at the origin.
Example 3.2.27
Extend f=x4+x2+x y2+y2x2−x2y+y2 to a function gx,y that is continuous at the origin.
Example 3.2.28
Prove Inequalities 1 and 2 in Table 3.2.1.
<< Previous Section Table of Contents Next Section >>
© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document