Limits of Functions of Several Variables

A limit is simply the value a function is heading toward as you get closer and closer to a point, whether or not it ever gets there. All we're adding now is a second input. The notation $\lim_{(x,y)\to(a,b)} f(x, y) = L$ looks heavy, but it just says: as the point $(x, y)$ (wherever you are on the map) closes in on $(a, b)$ (the target spot), the height $f(x, y)$ settles toward $L$. The only new wrinkle is that there are now infinitely many ways to walk toward the target, and they all have to agree.

Think back to one-variable limits. There, $\lim_{x\to a} f(x)$ exists when the left-hand and right-hand limits match: two directions, one check. In the plane you can walk toward $(a, b)$ from the east, from the northwest, along a parabola, even spiraling in. For the limit to exist, every one of those paths has to lead to the same height.

Continuous functions are the easy case, just like before. For polynomials, and sums, products and compositions of the usual functions wherever they're defined, you can substitute the point and you're done.

Showing a limit doesn't exist is the other easy case: find two paths that give different values. Straight lines $y = mx$ through the point are the first thing to try, since you can check all of them at once by keeping $m$ as a letter. If they all agree, try curves like $y = x^2$.

A common mistake. It's tempting to think that if every straight line gives the same value, the limit must exist, because lines seem to cover every direction. It isn't so: $\frac{x^2y}{x^4 + y^2}$ is $0$ along every line through the origin, but $\frac12$ along the parabola $y = x^2$. A curved path can sneak in a way no line does. Matching paths can only ever disprove a limit, never prove one.

Showing a limit does exist takes a different tool: one that handles all paths at once. Usually that's the squeeze theorem, or polar coordinates. Polar coordinates work nicely because $r$ is exactly the distance to the origin. If the expression goes to $0$ as $r \to 0$ no matter what $\theta$ is doing, then every path gives $0$, and the limit is $0$.

Numerical simulations rely on continuity: refining a grid should make results converge to one answer no matter how the grid is laid out.