Limits and Continuity in R^n

Part 3, Chapter 3: Calculus of Several Variables

Learning objectives

Extend the $\epsilon\text{-}\delta$ definition to multivariable functions using the Euclidean norm
Apply the two-path test to show a limit does not exist
Recognize that polynomials in several variables are continuous everywhere
Predict whether common multivariable limits exist by inspecting their dominant terms

Limits in $\mathbb{R}^n$ are harder than limits in $\mathbb{R}$ for one reason: in one variable you can only approach a point from the left or the right, but in two variables you can approach from infinitely many directions. A limit at $(0,0)$ has to be the same along every straight line, every parabola, every spiral, every conceivable path, or it does not exist. This single fact explains why so many "natural-looking" multivariable expressions fail to have limits at the origin. The cure is the same as in single-variable analysis: use the $\epsilon\text{-}\delta$ machinery, but with the Euclidean norm.

The definition

For $f:\mathbb{R}^n\to\mathbb{R}^m$ , we say $\lim_{\mathbf{x}\to\mathbf{a}}f(\mathbf{x})=\mathbf{L}$ mathbfxtomathbfaf(mathbfx)=mathbfL when: for every $\epsilon>0$ there is a $\delta>0$ such that whenever $0<\|\mathbf{x}-\mathbf{a}\|<\delta$ , we have $\|f(\mathbf{x})-\mathbf{L}\|<\epsilon$ . Here $\|\cdot\|$ is the Euclidean norm: $\|\mathbf{x}\|=\sqrt{x_1^2+\cdots+x_n^2}$ . The picture: every input inside an open ball of radius $\delta$ around $\mathbf{a}$ gets mapped to within an open ball of radius $\epsilon$ around $\mathbf{L}$ .

A function $f$ is continuous at $\mathbf{a}$ when $\lim_{\mathbf{x}\to\mathbf{a}} f(\mathbf{x})=f(\mathbf{a})$ mathbfxtomathbfaf(mathbfx)=f(mathbfa). Polynomials in several variables are continuous everywhere; rational functions are continuous wherever the denominator is non-zero. Sums, products, compositions, and quotients of continuous functions are continuous (where defined). These standard rules let you certify continuity of complicated expressions almost by inspection.

The two-path test (your most useful tool)

To prove a limit does NOT exist, find two paths to $\mathbf{a}$ along which $f$ approaches different values. The textbook example: $f(x,y)=\frac{xy}{x^2+y^2}$ at the origin. Along the $x$ -axis ( $y=0$ ): $f=0$ , so limit $=0$ . Along the line $y=x$ : $f=\frac{x^2}{2x^2}=1/2$ . Two different limits → no limit exists. This pattern shows up constantly: any ratio whose numerator and denominator are homogeneous polynomials of the same degree is prone to this.

The grapher above plots $y=f(x)$ for one variable. Try plotting $x^2/(x^2+0.01)$ for small offsets to see how a denominator going to zero produces wildly different limits depending on the rate. The same intuition extends to two variables: when both numerator and denominator vanish at the same rate, the limit depends on the direction of approach.

Where this shows up

Numerical optimization, ill-conditioning: The condition number of a function $f:\mathbb{R}^n\to\mathbb{R}$ near a saddle or ridge depends on directional derivatives that may not agree. Optimizers (Newton, conjugate gradient) probe along multiple directions for exactly the path-dependence reason described above.
Machine learning, loss-surface analysis: Saddle points in high-dimensional loss landscapes are characterized by direction-dependent second derivatives. Modern deep-learning optimizers (Adam, SGD with momentum) are designed to escape regions where the gradient's limit "behavior" disagrees along different directions.
PDE numerical methods: Adaptive mesh refinement detects regions where a solution's gradient is path-dependent (a shock or boundary layer) and refines the grid there. These are precisely where multivariable limits fail to be continuous.

Pause and think: The expression $\frac{x^2}{x^2+y^2}$ at $(0,0)$ . Along $y=0$ it equals 1. Along $x=0$ it equals 0. So the limit does not exist. Now think about $\frac{x^2 y}{x^2+y^2}$ . Try the same two paths. Why does this one have a limit but the first doesn't? (Hint: a factor of $y$ in the numerator forces the whole expression to zero faster than the denominator.)

Try it

Predict first: does $\lim_{(x,y)\to(0,0)}\frac{x^2-y^2}{x^2+y^2}$ (x,y)to(0,0)fracx2−y2x2+y2 exist? Test the $x$ -axis and the $y$ -axis.
Predict first: does $\lim_{(x,y)\to(0,0)}\frac{x^3+y^3}{x^2+y^2}$ (x,y)to(0,0)fracx3+y3x2+y2 exist? Convert to polar: $x=r\cos\theta,y=r\sin\theta$ . The expression becomes $r(\cos^3\theta+\sin^3\theta)\to 0$ .
Show that the squeeze theorem proves $\lim_{(x,y)\to(0,0)}\frac{x^2 y^2}{x^2+y^2}=0$ (x,y)to(0,0)fracx2y2x2+y2=0. Bound the expression by $y^2$ and let $y\to 0$ .
Predict: is $f(x,y)=3x^2+2xy-5y^3$ continuous at $(1,2)$ ? Why? (Polynomials are continuous everywhere, so compute $f(1,2)$ and you are done.)
Trap question: does the path test along $y=mx$ (every straight line through the origin) being independent of $m$ prove a limit exists? No, you also need parabolas, etc. Find a function whose limit is the same along every line but DNE along $y=x^2$ : try $\frac{x^2 y}{x^4+y^2}$ .

A trap to watch for

Showing the limit along every straight line through $(0,0)$ is the same value does NOT prove the limit exists. The function $f(x,y)=\frac{x^2 y}{x^4+y^2}$ is the canonical counterexample: along $y=mx$ the limit is 0 for every slope $m$ , but along the parabola $y=x^2$ the limit is $1/2$ . Curved paths can reveal hidden non-existence that straight-line tests miss. To prove a limit exists you almost always need the squeeze theorem or polar coordinates; straight-line agreement is suggestive but not sufficient.

What you now know

You can run the two-path test to disprove limits, use the squeeze theorem and polar coordinates to prove them, and recognize that polynomials and rational functions inherit continuity from their components. The next section turns to the multivariable derivative, the Jacobian, which inherits its meaning entirely from these limit definitions.

Mark section complete →

References

Garrity, T. (2002). All the Mathematics You Missed. Cambridge UP, ch. 3.
Spivak, M. (1965). Calculus on Manifolds. W. A. Benjamin, ch. 2.
Munkres, J. R. (1991). Analysis on Manifolds. Westview Press, ch. 1.
Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill, ch. 4 and 9.
Apostol, T. M. (1974). Mathematical Analysis (2nd ed.). Addison-Wesley, ch. 12.