The Main Idea 

Taking limits of functions with two variables follows similar rules to single-variable limits, but there’s a crucial difference: you can approach a point from infinitely many directions.

The Definition

For [latex]\underset{(x,y)\to(a,b)}{\text{lim}}f(x,y)=L[/latex] to exist, [latex]f(x,y)[/latex] must get arbitrarily close to [latex]L[/latex] as [latex](x,y)[/latex] approaches [latex](a,b)[/latex] from any direction.

We use a [latex]\delta[/latex] disk: a circle of radius [latex]\delta[/latex] around [latex](a,b)[/latex]. All points in this disk (except possibly [latex](a,b)[/latex] itself) must give [latex]f[/latex] values close to [latex]L[/latex].

The Limit Laws

The familiar limit laws extend to two variables:

• Sum, difference, product, quotient laws
• Constant multiple, power, root laws
• Identity laws: [latex]\displaystyle\lim_{(x,y)\to(a,b)}x=a[/latex] and [latex]\displaystyle\lim_{(x,y)\to(a,b)}y=b[/latex]

Use these laws to break down complex limits into simpler pieces.

When Limits Don’t Exist

The limit must be the same along every possible path. If two different paths give different values, the limit doesn’t exist.

Problem-Solving Strategy

Try substitution first using limit laws. If that fails, test different paths: try [latex]y=0[/latex], [latex]x=0[/latex], [latex]y=x[/latex], or curves like [latex]y=x^2[/latex]. If any two paths give different values, the limit doesn’t exist.

The Main Idea 

To understand continuity and limits in multivariable calculus, you need to distinguish between points inside a region and points on its edge.

A point [latex]P_0[/latex] is an interior point of set [latex]S[/latex] if you can draw a small disk around it that stays completely inside [latex]S[/latex]. Think of it as having breathing room—every nearby point is also in [latex]S[/latex].

A point [latex]P_0[/latex] is a boundary point of [latex]S[/latex] if every disk around it contains both points inside [latex]S[/latex] and points outside [latex]S[/latex]. It sits on the edge.

Open and Closed Sets

• Open set: Every point is an interior point (no boundary included)
• Closed set: Contains all its boundary points

A disk without its edge is open. A disk with its edge is closed. Include only part of the edge? Neither open nor closed.

Connected Sets and Regions

• Connected: Can’t be split into separate pieces
• Region: Open, connected, and nonempty

Limits at Boundary Points

When taking limits at boundary points, you only consider points inside both the [latex]\delta[/latex] disk and the domain. Some of the disk lies outside the domain—that’s fine. Just ignore those points.

The Main Idea 

A function of two variables is continuous at a point when there are no jumps, breaks, or holes at that point. The definition mirrors single-variable continuity.

The Three Conditions

A function [latex]f(x, y)[/latex] is continuous at [latex](a, b)[/latex] if:

1. [latex]f(a, b)[/latex] exists
2. [latex]\displaystyle\lim_{(x,y)\to(a,b)}f(x,y)[/latex] exists
3. [latex]\displaystyle\lim_{(x,y)\to(a,b)}f(x,y)=f(a,b)[/latex]

All three must hold. The function must be defined, the limit must exist, and they must equal each other.

Checking Continuity

Verify each condition:

• Check the point is in the domain
• Calculate the limit using limit laws
• Confirm the limit equals the function value

Properties of Continuous Functions

• Sum: If [latex]f[/latex] and [latex]g[/latex] are continuous, so is [latex]f + g[/latex]
• Product: If [latex]g(x)[/latex] and [latex]h(y)[/latex] are continuous, so is [latex]g(x)h(y)[/latex]
• Composition: If [latex]g(x, y)[/latex] is continuous and [latex]f(z)[/latex] is continuous, so is [latex]f(g(x, y))[/latex]

Common Continuous Functions

Polynomials, rational functions (where denominators aren’t zero), trig functions, exponentials, and logarithms are continuous on their domains. Compositions and combinations of these remain continuous.

The Main Idea 

Limits and continuity extend naturally beyond two variables. The key is replacing [latex]\delta[/latex] disks with [latex]\delta[/latex] balls in higher dimensions.

The [latex]\delta[/latex] Ball

In three dimensions, a [latex]\delta[/latex] ball around [latex](x_0, y_0, z_0)[/latex] includes all points within distance [latex]\delta[/latex]:

[latex]{(x, y, z) \mid \sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2} < \delta}[/latex]

For four dimensions, add another squared term under the radical. The pattern continues for any number of dimensions.

Limits in Three Variables

The definition mirrors two variables. For [latex]\displaystyle\lim_{(x,y,z)\to(a,b,c)}f(x,y,z)=L[/latex], points in a [latex]\delta[/latex] ball around [latex](a,b,c)[/latex] must give function values arbitrarily close to [latex]L[/latex].

Using Limit Laws

All limit laws from two variables work for three or more variables:

• Sum, difference, product, quotient
• Constant multiple, power, root
• Identity laws

Continuity

A function is continuous at a point if the same three conditions hold: the function exists, the limit exists, and they’re equal. The concept is identical—only the dimension changes.