How to Prove a Multivariable Limit with Epsilon and Delta

A limit of a function of two variables is proved by showing that points sufficiently close to $(a,b)$ make the function value sufficiently close to $L$. The distance is measured in the plane, usually with the Euclidean norm. The proof follows the same logic as a one-variable $\epsilon$-$\delta$ proof, but the input can approach the point from any direction.

The definition for two variables

The statement

$$\lim_{(x,y)\to(a,b)} f(x,y)=L$$

means that for every $\varepsilon>0$, there is a $\delta>0$ such that

$$0<\sqrt{(x-a)^2+(y-b)^2}<\delta \quad\Longrightarrow\quad |f(x,y)-L|<\varepsilon.$$

The quantity $\sqrt{(x-a)^2+(y-b)^2}$ is the Euclidean distance between $(x,y)$ and $(a,b)$. The condition that this distance is positive excludes the point $(a,b)$ itself; a limit concerns nearby points, not necessarily the function’s value at the point.

The order of the choices matters: you are given an arbitrary $\varepsilon>0$, and then you must specify a suitable $\delta$. Your choice may depend on $\varepsilon$, but not on the particular point $(x,y)$.

A practical proof method

Start by examining $|f(x,y)-L|$. Try to rewrite or estimate it in terms of $|x-a|$, $|y-b|$, or the combined distance

$$r=\sqrt{(x-a)^2+(y-b)^2}.$$

Useful inequalities include $|x-a|\le r$ and $|y-b|\le r$. If you can show that $|f(x,y)-L|\le Cr$ near $(a,b)$ for some constant $C>0$, then choosing $\delta=\varepsilon/C$ is often enough. If your estimate only works in a restricted neighborhood, also require $\delta$ to be small enough to stay in that neighborhood.

A typical proof therefore has four parts: let $\varepsilon>0$ be arbitrary; choose $\delta$; assume $0<r<\delta$; and use the estimate to prove $|f(x,y)-L|<\varepsilon$.

Worked example: a limit at a nonzero point

Prove that

$$\lim_{(x,y)\to(1,2)}(x^2+y^2)=5.$$

Let $\varepsilon>0$. Factor the difference from the proposed limit:

$$\begin{aligned} |x^2+y^2-5| &=|(x^2-1)+(y^2-4)|\\ &\le |x-1||x+1|+|y-2||y+2|. \end{aligned}$$

We first restrict the points to be within distance $1$ of $(1,2)$. Then $|x-1|<1$ and $|y-2|<1$, so $0<x<2$ and $1<y<3$. Consequently, $|x+1|<3$ and $|y+2|<5$. Thus

$$|x^2+y^2-5|<3|x-1|+5|y-2|.$$

Each coordinate difference is no greater than the Euclidean distance $r=\sqrt{(x-1)^2+(y-2)^2}$. Therefore,

$$|x^2+y^2-5|<8r.$$

Choose

$$\delta=\min\left(1,\frac{\varepsilon}{8}\right).$$

If $0<r<\delta$, then the estimates above apply and

$$|x^2+y^2-5|<8r<8\delta\le\varepsilon.$$

This proves the limit. The minimum in the choice of $\delta$ handles two requirements at once: staying within distance $1$ for the bounds, and making the final error less than $\varepsilon$.

Using paths to test a proposed limit

A limit must work along every way of approaching $(a,b)$. If two paths approaching the point give different limiting function values, the multivariable limit does not exist. For example, consider

$$f(x,y)=\frac{xy}{x^2+y^2},\qquad (x,y)\ne(0,0).$$

Along $y=x$, the function is $1/2$ for nonzero $x$. Along $y=0$, it is $0$. Since these approach different values as $(x,y)\to(0,0)$, the limit does not exist. Path checks can disprove a limit, but checking a few paths cannot prove one: a successful epsilon-delta argument must cover all nearby points.

Common mistakes

• Using separate coordinate conditions without connecting them to the definition. You may work with $|x-a|$ and $|y-b|$, but explain how the distance condition controls them. If $r<\delta$, then both coordinate differences are less than $\delta$.
• Choosing $\delta$ before seeing the needed bound. Derive an estimate for $|f(x,y)-L|$ first, then choose $\delta$ to make that estimate less than $\varepsilon$.
• Forgetting local restrictions. Factoring expressions often produces terms such as $|x+a|$ that need to be bounded. Restricting $r<1$ (or another convenient radius) can provide those bounds; include that restriction in the choice of $\delta$.
• Treating path checks as a proof of existence. Agreement along several paths is not enough to control every direction or curved approach.
• Requiring the function to equal the limit at the point. The definition uses nearby points with positive distance from $(a,b)$, so $f(a,b)$ may be different or undefined.