Limits and Continuity in ℝⁿ

A limit at a point in one variable requires agreement from only two directions (left and right). In $\mathbb{R}^2$, approaching a point along infinitely many paths all gives a potential limiting value, and they must all agree. This extra richness is not merely a technicality: the function $f(x,y) = xy/(x^2+y^2)$ has limit 0 along every axis but limit $1/2$ along the diagonal — so its limit at the origin does not exist. The failure of path-independence is the primary tool for proving a limit does not exist in several variables, and it is also the source of the subtlety in differentiability (next lesson). Continuity at a point — as in one dimension — simply requires the limit to equal the actual value.

• state the $\varepsilon$-$\delta$ definition of $\lim_{(x,y)\to(x_0,y_0)} f(x,y) = L$;
• use the two-path test to show that a limit does not exist;
• verify a limit using the $\varepsilon$-$\delta$ definition or by recognising it as a polynomial/rational composition;
• state the definition of continuity at a point and identify where standard functions are continuous;
• apply limit laws (sum, product, quotient, composition) for multivariable functions.

1. 1. The ε-δ Definition
2. 1. The Two-Path Test
3. 1. Continuity
4. 1. ε-δ Proofs — The Strategy