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Chapter 3: Functions of Several Variables
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Section 3.2: Limits and Continuity
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Essentials



Limit



Definition of the Bivariate Limit


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It is first necessary to distinguish between interior and boundary points of the domain of a function $f$.

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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 $f$.

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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.

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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:$\left(a\,b\right)$ is an interior point of the domain of $f$, then the number $L$ is the bivariate limit of $f\left(x\,y\right)$ at P, that is, $\underset{\left(x\,y\right)\to P}{lim}f\left(xcomma;y\right)equals;L$, when, for every number $\mathrm{\ε}\>0$ there is a corresponding number $\mathrm{\δ}$ with the property that
$0<\sqrt{{\left(xa\right)}^{2}\+{\left(yb\right)}^{2}}<\mathrm{\δ}$ ⇒$\leftf\left(x\,y\right)L\right<\mathrm{\ε}$
If P:$\left(a\,b\right)$ is a boundary point of the domain of $f$, then $\underset{\left(x\,y\right)\to P}{lim}f\left(xcomma;y\right)equals;L$, when, for every number $\mathrm{\ε}\>0$ there is a corresponding number $\mathrm{\δ}$ with the property that
$0<\sqrt{{\left(xa\right)}^{2}\+{\left(yb\right)}^{2}}<\mathrm{\δ}$ and $\left(x\,y\right)$ is in the domain of $f$ ⇒$\leftf\left(x\,y\right)L\right<\mathrm{\ε}$



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The points satisfying $0<\sqrt{{\left(xa\right)}^{2}\+{\left(yb\right)}^{2}}<\mathrm{\delta}$ are said to lie in a deleted neighborhood of $\left(a\,b\right)$. This deleted neighborhood is actually the interior of an annulus that is an open disk of radius $\mathrm{\δ}$ with the center point $\left(a\,b\right)$ removed.

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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 $\left(x\,y\right)$ sufficiently close to P. Thus, $L$ is the limit of $f\left(x\,y\right)$ at P if all the points in the deleted neighborhood of P produce function values that are close to $L$.

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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


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It is generally difficult to prove that $L$ is the limiting value of $f\left(x\,y\right)$ because the requisite estimates demand a facility with manipulating inequalities.

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Table 3.2.1 lists several inequalities that are useful for proving that a real number $L$ is indeed the bivariate limit of $f\left(x\,y\right)$. Inequality 3 is the "triangle" inequality. Inequalities 4, 5, 6, and 7 should be selfevident. A proof of Inequality 2 uses Inequality 1, which itself is proved in Example 3.2.28.

Reference

Inequality

Inequality 1

$\leftxy\right$ $\le \left({x}^{2}\+{y}^{2}\right)\/2$

Inequality 2

$\x\\+\lefty\right$ $\le \sqrt{2}\sqrt{{x}^{2}plus;{y}^{2}}$${}$

Inequality 3

$\x\+y\\le \leftx\right\+\lefty\right$

Inequality 4

$\leftx\right\le \sqrt{{x}^{2}\+{y}^{2}}$

Inequality 5

$\lefty\right\le \sqrt{{x}^{2}\+{y}^{2}}$

Inequality 6

${x}^{2}\le {x}^{2}\+{y}^{2}$

Inequality 7

${y}^{2}\le {x}^{2}\+{y}^{2}$

Table 3.2.1 Useful inequalities



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The Bivariate Limit in Maple


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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).
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Showing the Limit Does Not Exist


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It is far easier to show that $f\left(x\,y\right)$ does not have a limit at $\left(a\,b\right)$. Recall that for a limit on the line to exist, both lefthand and righthand 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 $\left(a\,b\right)$, it suffices to show that along two different paths through $\left(a\,b\right)$ the limits differ, or that one such limit does not exist.

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Typical trial paths are the axes, lines $y\=mx$, parabolas $y\={x}^{2}$ and $x\={y}^{2}$, and on rare occasions, the curves $y\={x}^{3\/2}$ and $y\={x}^{2\/3}$.




Continuity


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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:$\left(a\,b\right)$ if $\underset{\left(x\,y\right)\to P}{lim}fequals;f\left(acomma;b\right)$.



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If $f$ is continuous at every point in its domain, then $f$ is said to be a continuous function on that domain.

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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

1.

$f\left(x\,y\right)$ is continuous at P:$\left(a\,b\right)$

2.

$g\left(x\right)$ is continuous at $f\left(a\,b\right)$

3.

$h\=g\circ f$ so that $h\left(x\,y\right)\=g\left(f\left(x\,y\right)\right)$

⇒
1.

$h$ is continuous at P







Examples


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Let P be the generic point $\left(x\,y\right)$ and O, the origin, $\left(0\,0\right)$.
For each $f$ in Examples 3.2.(110), show that $\underset{\mathrm{P}\to \mathrm{O}}{\mathrm{lim}}f\left(x\,y\right)$, the bivariate limit at the origin, does not exist.
Example 3.2.1

$f\=\frac{{x}^{2}}{{x}^{2}\+{y}^{2}}$


Example 3.2.6

$f\=\frac{xy}{x\+y}$

Example 3.2.2

$f\=\frac{{x}^{2}{y}^{2}}{{x}^{2}\+{y}^{2}}$

Example 3.2.7

$f\=\frac{xy}{{x}^{4}plus;{y}^{4}}$

Example 3.2.3

$f\=\frac{xy}{{x}^{2}\+{y}^{2}}$

Example 3.2.8

$f\=\frac{{x}^{3}{y}^{2}}{{x}^{6}\+{y}^{4}}$

Example 3.2.4

$f\=\frac{x{y}^{2}}{{x}^{2}plus;{y}^{4}}$

Example 3.2.9

$f\=\frac{{x}^{4}\+2{x}^{2}{y}^{2}plus;3x{y}^{3}}{{\left({x}^{2}plus;{y}^{2}\right)}^{2}}$

Example 3.2.5

$f\=\frac{{x}^{2}\+y}{\sqrt{{x}^{2}\+{y}^{2}}}$

Example 3.2.10

$f\=\frac{{x}^{4}{y}^{4}}{{\left({x}^{2}\+{y}^{4}\right)}^{3}}$



Example 3.2.11

Show that for $f\=\frac{xy}{{x}^{2}plus;{y}^{2}}$ the bivariate limit at the origin does not exist, but the iterated limits $\underset{y\to 0}{lim}\left(\underset{x\to 0}{lim}f\right)$ and $\underset{x\to 0}{lim}\left(\underset{y\to 0}{lim}f\right)$ are both zero.

Example 3.2.12

Show that for $f\=\frac{xy}{{x}^{3}plus;{y}^{2}}$ the bivariate limit at the origin does not exist, but the iterated limits $\underset{y\to 0}{lim}\left(\underset{x\to 0}{lim}f\right)$ and $\underset{x\to 0}{lim}\left(\underset{y\to 0}{lim}f\right)$ are both zero.

Example 3.2.13

Prove that the bivariate limit at the origin for $f\=\frac{{x}^{3}}{{x}^{2}\+{y}^{2}}$ is zero.

Example 3.2.14

Prove that the bivariate limit at the origin for $f\=\frac{2{x}^{3}{y}^{3}}{{x}^{2}plus;{y}^{2}}$ is zero.

Example 3.2.15

Prove that the bivariate limit at the origin for $f\=xy\frac{{x}^{2}{y}^{2}}{{x}^{2}plus;{y}^{2}}$ is zero.

Example 3.2.16

Prove that the bivariate limit at the origin for $f\=\frac{{x}^{2}{y}^{2}}{{x}^{2}\+{y}^{2}}$ is zero.

Example 3.2.17

Prove that the bivariate limit at the origin for $f\=\frac{{x}^{4}\+{y}^{4}}{{x}^{2}\+{y}^{2}}$ is zero.

Example 3.2.18

Prove that the bivariate limit at the origin for $f\=\frac{2{x}^{5}plus;2{y}^{3}\left(2{x}^{2}{y}^{2}\right)}{{\left({x}^{2}plus;{y}^{2}\right)}^{2}}$ is zero.

Example 3.2.19

If $f\left(x\,y\right)\=2{x}^{2}6xyplus;5{y}^{2}comma;$prove that $\sqrt{{x}^{2}\+{y}^{2}}<\sqrt{\mathrm{\ε}\/8}\=\mathrm{\δ}$ ⇒ $\f\left(x\,y\right)\<\mathrm{\ε}$.

Example 3.2.20

Prove the inequality $\left{x}^{3}{y}^{3}\right\le {\left({x}^{2}\+{y}^{2}\right)}^{3\/2}$.

Example 3.2.21

Show that for $f\=x\mathrm{sin}\left(1sol;y\right)plus;y\mathrm{sin}\left(1sol;x\right)$ the bivariate limit at the origin is zero, but both of the iterated limits $\underset{y\to 0}{lim}\left(\underset{x\to 0}{lim}f\right)$ and $\underset{x\to 0}{lim}\left(\underset{y\to 0}{lim}f\right)$ fail to exist.
Hint: Show $\leftf\right$$\le \leftx\right\+\lefty\right\le 2\sqrt{{x}^{2}\+{y}^{2}}$.

Example 3.2.22

Show that for $f\=\frac{xy}{{x}^{2}plus;{y}^{2}}plus;x\mathrm{sin}\left(1sol;y\right)$ the bivariate limit at the origin and the iterated limit $\underset{x\to 0}{lim}\left(\underset{y\to 0}{lim}f\right)$ both fail to exist, but the iterated limit $\underset{y\to 0}{lim}\left(\underset{x\to 0}{lim}f\right)$ is zero.

Example 3.2.23

Show that the bivariate limit at the origin for $f\=\{\begin{array}{cc}\frac{\leftx\right}{{y}^{2}}{e}^{\x\\/{y}^{2}}& y\ne 0\\ 0& y\=0\end{array}$ does not exist.

Example 3.2.24

Extend $f\=\frac{{x}^{2}y\mathrm{cos}\left(xy\right)}{{x}^{2}\+{y}^{2}}$ to a function $g\left(x\,y\right)$ that is continuous at the origin.

Example 3.2.25

Extend $f\=\frac{\mathrm{tan}\left(xy\right)}{\mathrm{tan}\left(x\right)\mathrm{tan}\left(y\right)}$ to a function $g\left(x\,y\right)$ that is continuous at the origin.

Example 3.2.26

Extend $f\=\frac{x\mathrm{sin}\left(xy\right)}{2\mathrm{cos}\left(x\right)\mathrm{cos}\left(y\right)}$ to a function $g\left(x\,y\right)$ that is continuous at the origin.

Example 3.2.27

Extend $f\=\frac{{x}^{4}\+{x}^{2}\+x{y}^{2}plus;{y}^{2}}{{x}^{2}{x}^{2}yplus;{y}^{2}}$ to a function $g\left(x\,y\right)$ that is continuous at the origin.

Example 3.2.28

Prove Inequalities 1 and 2 in Table 3.2.1.



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