Limits in Multiple Variables

Calculus 3 · Axiom Academy

LESSON Limits in Multiple Variables How limits work in — and why infinitely many approach paths make them far subtler than in one variable. The one-variable definition survives almost word for word. The only thing that changes is how we measure closeness : in the distance from (x,y) to (a,b) is the Euclidean distance, so a single pins down a whole disk around the point — not an interval with two ends. For f(x,y) = 2x+y at (1,1) , the Cauchy–Schwarz bound hands you directly. In one dimension there are only two ways in: from the left and from the right. In two dimensions there are infinitely many . Watch what happens to as a point slides toward the origin along a straight line: the value never budges — but change the line, and it lands somewhere else. Every line gives its own answer Substituting y = mx kills the x entirely — the value depends only on the slope. So m = 0 gives 0 , m = 1 gives , m = 2 gives , and m = 3 gives . Four paths, four answers. Note also that y = 0 and x = 0 both return 0 . Two paths agreeing is not evidence of anything — it only takes one disagreement to settle the question. 3. Polar Coordinates Technique When the point of interest is the origin, converting to polar coordinates is often the cleanest move. Approaching (0,0) from any direction whatsoever becomes the single statement , and the direction is carried by . The r factors out and the leftover part is bounded, so a single g(r) = r squeezes the whole ring at once:

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