Limit of a function of several variables

In one variable there are only two directions to approach \(a\): left and right. With two variables there are infinitely many paths toward \((a,b)\): lines, parabolas, spirals. For the limit to exist, \(f\) must approach the same \(L\) along all of them. Formally, for every \(\varepsilon>0\) there must be a \(\delta>0\) such that \(|f(\mathbf x)-L|<\varepsilon\) whenever \(0<\|\mathbf x-\mathbf a\|<\delta\).

This gives two practical rules. If \(f\) is a polynomial, a rational function with nonzero denominator or a composition of continuous functions, the limit is found by substitution. If two paths give different values, the limit does not exist; for example, \(\frac{xy}{x^2+y^2}\) is \(0\) on the \(x\)-axis and \(\frac12\) on \(y=x\). But beware: testing many paths never proves that the limit exists, as \(\frac{xy^2}{x^2+y^4}\) shows, giving \(0\) along every line and \(\frac12\) along \(x=y^2\).

To prove existence one bounds the expression, with the squeeze theorem, or switches to polar coordinates: with \(x=r\cos\theta\), \(y=r\sin\theta\), it suffices that the expression tends to \(L\) as \(r\to0\) regardless of \(\theta\). The one-variable limit laws (sum, product, quotient) still hold, and from this concept follow continuity and, with it, partial derivatives.