Limits and Continuity in Multivariable Calculus

This study guide covers the fundamental concepts of limits and continuity for functions of multiple variables, specifically focusing on functions of the form $z = f(x, y)$. Unlike single-variable calculus, where a limit is approached from only two directions (left and right), multivariable limits must be consistent across an infinite number of paths.

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Learning Objectives

After studying this material, you should be able to:

• Define the limit of a function of two variables using the $\epsilon-\delta$ formal definition.
• Understand the concept of an open disk as the multivariable equivalent of an open interval.
• Evaluate limits of multivariable functions using algebraic limit laws.
• Prove that a limit does not exist using the Two-Path Test.
• Verify the continuity of a function at a specific point $(x_0, y_0)$.

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The "Big Idea"

In single-variable calculus, the limit exists if the left-hand limit equals the right-hand limit. In multivariable calculus, the domain is a plane (or higher-dimensional space). For a limit $\lim_{(x,y) \to (x_0,y_0)} f(x,y) = L$ to exist, the function must approach $L$ regardless of which path in the $xy$-plane you take to reach $(x_0, y_0)$. If you find even two paths that yield different results, the limit does not exist (DNE).

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Key Terms & Glossary

• Open Disk: A set of points $(x, y)$ such that the distance to a center point $(a, b)$ is less than a radius $\delta$. Formally: $\sqrt{(x-a)^2 + (y-b)^2} < \delta$.
  • Example: The region inside a circle, not including the boundary line.
• $\epsilon-\delta$ Definition: The formal proof framework for limits. For every error tolerance $\epsilon > 0$, there exists a distance $\delta > 0$ such that if the input is within $\delta$ of the target point, the output is within $\epsilon$ of the limit $L$.
• Path-Dependence: A property where the limit value changes based on the curve (path) followed to reach the point.
  • Example: Approaching $(0,0)$ along the x-axis vs. the line $y=x$.
• Continuity: A function $f(x, y)$ is continuous at $(x_0, y_0)$ if the limit exists, the function is defined at that point, and the limit equals the function value.

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Formula / Concept Box

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Hierarchical Outline

• I. Limits of Functions of Two Variables
  • A. From Intervals to Disks: In 1D we use $|x-a| < \delta$; in 2D we use the distance formula for a disk.
  • B. The Formal Definition: Relates the proximity in the domain (disk) to the proximity in the range (z-axis).
• II. Evaluating Limits
  • A. Direct Substitution: Applicable for polynomials and rational functions where the denominator is non-zero.
  • B. Limit Laws: Sum, Difference, Constant Multiple, Product, Quotient, Power, and Root laws.
• III. Non-Existence of Limits
  • A. The Two-Path Test: If $f(x,y) \to L_1$ along path $C_1$ and $f(x,y) \to L_2$ along path $C_2$, and $L_1 \neq L_2$, then the limit DNE.
  • B. Common Paths to Test: $x=0$ (y-axis), $y=0$ (x-axis), $y=mx$ (lines), $y=x^2$ (parabolas).
• IV. Continuity
  • A. Three Requirements: $f(a,b)$ exists; $\lim_{(x,y) \to (a,b)} f(x,y)$ exists; they are equal.
  • B. Composition of Continuous Functions: If $g$ and $h$ are continuous, $f(x,y) = g(h(x,y))$ is continuous.

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Visual Anchors

Comparison: 1D vs 2D Approach

Visualization of the Delta-Disk

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Definition-Example Pairs

Definition: The Two-Path Test If a function $f(x, y)$ approaches different values along two different paths as $(x, y)$ approaches $(x_0, y_0)$, then the limit of $f(x, y)$ at that point does not exist.

[!TIP] Real-World Example: Think of a mountain pass. If you approach a peak from the North ridge, you might reach an altitude of 5,000ft. If you approach from the South valley, you might end at a cliff face at 4,800ft. Because the elevations don't meet at the same point from different directions, there is a "discontinuity" or no single "limit" height for that specific longitude/latitude coordinate.

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Worked Examples

Example 1: Proving a Limit Does Not Exist

Problem: Show that $\lim_{(x,y) \to (0,0)} \frac{x^2 - y^2}{x^2 + y^2}$ does not exist.

Step 1: Test Path 1 (x-axis, $y=0$) $\lim_{(x,0) \to (0,0)} \frac{x^2 - 0^2}{x^2 + 0^2} = \lim_{x \to 0} \frac{x^2}{x^2} = 1$

Step 2: Test Path 2 (y-axis, $x=0$) $\lim_{(0,y) \to (0,0)} \frac{0^2 - y^2}{0^2 + y^2} = \lim_{y \to 0} \frac{-y^2}{y^2} = -1$

Conclusion: Since $$1 \neq -1$$, the limit depends on the path and therefore does not exist.

Example 2: Evaluating via Limit Laws

Problem: Evaluate $\lim_{(x,y) \to (1,2)} (3x^2y - x + 5)$.

Solution: Because this is a polynomial, it is continuous everywhere. We can use direct substitution:

1. $3(1)^2(2) - (1) + 5$
2. $3(2) - 1 + 5$
3. $6 - 1 + 5 = 10$ The limit is 10.

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Checkpoint Questions

1. What is the geometric difference between a neighborhood in 1D calculus and 2D calculus?
2. If $\lim_{(x,y) \to (0,0)} f(x,y) = 5$ along the path $y=x$ and along the path $y=2x$, does this prove the limit is 5? Why or why not?
3. State the three conditions required for $f(x,y)$ to be continuous at $(a,b)$.

▶Click to expand answers

1. In 1D, it is an open interval $(a-\delta, a+\delta)$. In 2D, it is an open disk.
2. No. Testing paths can only prove a limit does not exist. To prove it does exist, you must use limit laws, squeeze theorem, or the $\epsilon-\delta$ definition, as there are infinitely many paths to check.
3. (1) $f(a,b)$ is defined; (2) The limit as $(x,y) \to (a,b)$ exists; (3) The limit equals $f(a,b)$.