Math 1C with Jeff Anderson
Part 1: Multidimensional Differential Calculus
Lesson 8: Limits and Continuity
©Jeffrey A. Anderson

Key Terms List: Briggs Section 12.3 p. 885
□  Limit of f(x,y) as (x,y) approaches (a,b) (p. 885)
□  Definition: Precise definition of Limit of f(x,y) as (x,y) approaches (a,b) (p. 886)
□  Theorem 12.1: Limits of Constant and Linear Functions (p. 886)
□  Theorem 12.2: Limit Laws for Functions of Two Variables (p. 887)
□  Procedure: Two-Path Test for Nonexistence of Limits (p. 890)
□  Definition of continuity on a domain for multivariable function (p. 890)
□  Theorem 12.3: Continuity of Composite Functions (p. 891)

Examples of Finding Limits for Multivariable Functions

Example 12.3.1 p. 887: Existent Limit

Using Brigg’s text, we see that the function

f(x,y) =3(x^2)y + (Sqrt[xy])

does not have a limit as (x,y) ⟶(0,0). In order to substantiate this claim, let’s begin by graphing the function on a limited domain that includes the point (0,0).

First, let’s initialize the function:

f[x_,y_]:=3(x^2)y + (Sqrt[x*y]);

Next, let’s use Plot3D to visualize the surface corresponding to this function description. In this case, we will let -5 ≤ x ≤ 5 and -5 ≤ y ≤ 5. We will also set our bounds for our output to be between -4 and 4.

Manipulate[
GraphicsRow[
{Show[
	ContourPlot[
		(((x-2))^2)+(((y-8))^2)==(δ^2),
		{x,0,4},
		{y,6,10}
	],
	Graphics[{PointSize[.02],Point[{2,8}]}],
	ImageSize-> {200,200}
],
Show[
		Plot3D[
		f[x,y],
		{x,0,4},
		{y,6,10},
		PlotRange-> {0,300},
		AxesLabel->{x,y,z},
		PlotStyle->Directive[Red,Specularity[White,20],Opacity[0.2]]
		],
		ParametricPlot3D[ 
			{2+δ*Sin[t],8+δ*Cos[t],f[2+δ*Sin[t],8+δ*Cos[t]]},
			{t, 0, 2π},
			PlotStyle->Directive[Blue,Thickness[0.01]]
		],		
		Graphics3D[{PointSize[.02],Point[{2,8,100}]}],
		ImageSize-> {300,300}
	]
}
],
	{{δ, 1}, 0, 1, 0.05}
]

Notice that there is something very interesting going on at the origin. Depending on which way we approach the origin, the surface depends differently.

We begin by first moving toward the origin along the x-axis. Because, Mathematica does not have a multivariable limit command, we combine two instances of the Limit command. In this case, since we want to move along the x-axis, we first limit our scope to the y = 0 position. Then, we follow the function along the x-axis toward zero.

Limit[Limit[f[x,y],y->8],x->2]

Next we change our direction. We now move toward the origin along the y-axis. Again we utilize two instances of the Limit command. Here we start by setting x = 0 and then moving toward y = 0.

Limit[Limit[f[x,y],x->2],y->8]

We notice that these limits yield different results. However, by our definition of a multivariable limit, we know that for a limit to exist, these values should be identical. Thus, we conclude that this limit DOES NOT EXIST.

Similar to Exercise 12.3.9 p. 750 - 751: Non-Existent Limit

Using Stewart’s text, we see that the function

f(x,y) = (((x^2)-(y^2))/((x^2)+(y^2)))

does not have a limit as (x,y) ⟶(0,0). In order to substantiate this claim, let’s begin by graphing the function on a limited domain that includes the point (0,0).

First, let’s initialize the function:

f[x_,y_]:=(((x^2)-(y^2))/((x^2)+(y^2)));

Next, let’s use Plot3D to visualize the surface corresponding to this function description. In this case, we will let -5 ≤ x ≤ 5 and -5 ≤ y ≤ 5. We will also set our bounds for our output to be between -4 and 4.

Plot3D[f[x,y],{x,-5,5},{y,-5,5},PlotRange-> {-2,2},AxesLabel->{x,y,z}]

Notice that there is something very interesting going on at the origin. Depending on which way we approach the origin, the surface depends differently.

We begin by first moving toward the origin along the x-axis. Because, Mathematica does not have a multivariable limit command, we combine two instances of the Limit command. In this case, since we want to move along the x-axis, we first limit our scope to the y = 0 position. Then, we follow the function along the x-axis toward zero.

Limit[Limit[f[x,y],y->0],x->0]

Next we change our direction. We now move toward the origin along the y-axis. Again we utilize two instances of the Limit command. Here we start by setting x = 0 and then moving toward y = 0.

Limit[Limit[f[x,y],x->0],y->0]

We notice that these limits yield different results. However, by our definition of a multivariable limit, we know that for a limit to exist, these values should be identical. Thus, we conclude that this limit DOES NOT EXIST.

Example 12.3.4 p. 890: Existent Limit

f[x_,y_]:=((x*(y^2))/((x^2)+(y^4)));
Plot3D[
	f[x,y],
	{x,-2,2},
	{y,-2,2},
	PlotRange-> {-1,1},
	AxesLabel->{x,y,z},
	BoxRatios->{1, 1, 1},
	ImageSize->{400,400}
]