Module 35: Multivariable functions, level sets, and limits

A. Prerequisites and observable goals#

B. The central idea: every nearby direction must agree#

A function $f(x,y)$ takes an ordered pair as input and returns one scalar. Its graph is a surface $z=f(x,y)$ in three dimensions, while its domain is a region in the input plane. A level curve $f(x,y)=c$ lives in that input plane; it marks inputs with a common output height. Confusing the domain, graph, and level set mixes objects of different dimensions.

Approaching a point now permits infinitely many paths. Different path limits disprove a common limit. Agreement along two paths, ten paths, or even every straight line does not by itself prove a multivariable limit; a curved approach may reveal different behavior.

C. The limit definition and useful bounds#

For a domain point approached from within the domain,

means that for every $\varepsilon>0$ there is $\delta>0$ such that $0<\sqrt{(x-a)^2+(y-b)^2}<\delta$ implies $|f(x,y)-L|<\varepsilon$. Continuity means the limit equals the point value. Standard sums, products, and compositions preserve continuity where they are defined, and a quotient is continuous where its denominator is nonzero. [O3Limits]

Let $r=\sqrt{x^2+y^2}$. Then $|x|\le r$, $|y|\le r$, and $2|xy|\le x^2+y^2=r^2$. Such bounds prove behavior uniformly in direction. In polar coordinates, $x=r\cos\theta,y=r\sin\theta$; a bound independent of $\theta$ that tends to zero as $r\to0$ can prove a limit. Checking each fixed angle separately is only checking straight rays, not all approaches where $\theta$ may change with $r$.

D. Why two-dimensional limits need stronger evidence#

For $f(x,y)=xy/(x^2+y^2)$ away from the origin, the axes give zero, but $y=x$ gives $1/2$. Thus no limit exists. A graph may look smooth from one chosen viewing direction while hiding this mismatch.

A more subtle example is $g=x^2y/(x^4+y^2)$. Along $y=mx$ with fixed nonzero $m$, the expression becomes $mx/(x^2+m^2)\to0$; along $y=0$ it is also zero. Yet along $y=x^2$ it is $1/2$. The denominator assigns different scales to the coordinates, suggesting the curved path. This is why an all-directions inequality is stronger than a collection of samples.

E. A decision procedure#

Determine the domain first. At a regular input, use continuity and direct substitution. At an indeterminate point, try simple paths to look for disagreement; inspect the powers in the denominator to select more informative curved paths. If evidence suggests a common limit, seek an inequality in $r$, an algebraic simplification valid on the punctured domain, or a uniform polar estimate. State whether you proved a limit or merely failed to find a counterpath.

F. Worked examples#

A domain with two restrictions. For $f(x,y)=\ln(y-x^2)/\sqrt{9-x^2-y^2}$, the logarithm requires $y>x^2$, while the denominator requires $x^2+y^2<9$, strictly. The domain is the portion above the parabola and inside the open disk satisfying both restrictions. A boundary satisfying only one is not included.

A successful uniform bound. For $h(x,y)=x^2y/(x^2+y^2)$ away from zero,

Therefore the limit is zero along every allowable approach. Defining $h(0,0)=0$ makes it continuous there. This proof does not assert differentiability, which is a stronger property addressed next.

Level curves as a height map. For $f(x,y)=x^2+4y^2$, the level $c>0$ is an ellipse with semiaxes $\sqrt c$ and $\sqrt c/2$. The level zero is one point, and negative levels are empty. Closely spaced level curves suggest rapid change, but precise rates require derivatives.

G. Independent practice#

F foundation C standard course work T transfer S optional stretch L runnable lab or project. Attempt the complete set before opening its solutions.

H. Complete solutions: Module 35#

Compare the first step where your work differs. A corrected answer is not yet an independent success.

Open every solution

I. Retention and readiness#

Retrieve M06.07 and M07.06. Explain the logical asymmetry: two disagreeing paths disprove a limit, but agreeing paths do not generally prove one. Advance when you can produce a uniform bound or identify precisely why your current evidence remains incomplete.