Professor Leonard – Calculus 3 – 13.2: The Limits of Continuity of Multivariable Functions (with Squeeze Theorem)
Thomas – 14.2 Limits of Continuity in Higher Dimensions
Why Can’t We Treat Limits of Multivariable Functions the Same as Single Variable Functions
Single-variable limit: \lim_{x \to a} f(x) = L
Multivariable limit: \lim_{(x,y) \to (a,b)} f(x,y) = L
In single-variable calculus, we write:
\lim_{x \to a} f(x) = L.
This means that as x approaches a, the function values f(x) approach L.
In multivariable calculus, we write:
\lim_{(x,y) \to (a,b)} f(x,y) = L.
This means that as the point (x,y) approaches the point (a,b), the function values f(x,y) approach L.
The limit is one of the first concepts encountered in Calculus. We can think of it as a function’s output value as the input value draws close to a specific value. We visualize the graph of a single-variable function as a curve upon which we approach a limit from the left and right. Here in Multivariable Calculus, we work with more than one input variable, and thus our graph takes on additional dimensions. In actuality, can only graph a two-variable function, given that more than that would demand work in four or more dimensions. So for the purposes of this blog entry, we’ll limit our discussion to two-variable functions.
The graph of a two-variable function is a surface, like a sheet manipulated into any conceivable formation. Two input values should produce a point on that surface. However, unlike the curve in two-dimensional space corresponding to our single-variable functions, a point on a surface can be approached from infinitely many paths. As such, for a limit to exist, the function must approach the same value along every possible path to the point. While it may in fact do so, proving that limit exists along infinitely many paths proves a significant step up in workload from the single-variable function.
Proving a Limit Doesn’t Exist
Rather than undertake the mission of proving a limit along infinitely many paths, we can take the path of least resistance by proving that a multivariable limit does not exist. To show that a limit doesn’t exist, we only need to show that two different paths to the same point produce different limit values. Since such a function approaches different values depending on the path, there is no single limit.
There isn’t a clear set of steps to prove that a limit doesn’t exist, although we learn some easy first steps. A good strategy is to try simple paths first. And what could be simpler than setting our x or y variables to zero and evaluating the limit of the non-zero independent variable?
Example
Consider the limit
\lim_{(x,y)\to(0,0)}\frac{x^2-y^2}{x^2+y^2}.
Approaching along the x-axis (y=0),
\frac{x^2-0}{x^2+0} =\frac{x^2}{x^2} =1.
Therefore,
\lim_{x\to0}f(x,0)=1.
Approaching along the y-axis (x=0),
\frac{0-y^2}{0+y^2} =\frac{-y^2}{y^2} =-1.
Therefore,
\lim_{y\to0}f(0,y)=-1.
Since the function approaches different values along the two paths,
\lim_{(x,y)\to(0,0)} \frac{x^2-y^2}{x^2+y^2} \text{ does not exist.}
However, if that first attempt yields equivalent limits, our job as persistent mathematicians isn’t finished. We can then try more creative and interesting paths, such as setting y = x, or x = y, or setting the variables to exponentials of each other. We could choose a path that matches the highest powers in the numerator and denominator.
Example
Consider the function
f(x,y)=\frac{x^2y}{x^4+y^2}.
First, try the simplest paths.
Along the x-axis (y=0),
[ f(x,0)=\frac{x^2(0)}{x^4+0^2}=0.
Along the y-axis (x=0),
f(0,y)=\frac{0^2y}{0^4+y^2}=0.
Both paths suggest that the limit might be 0.
Now try the path y=x^2.
f(x,x^2) =\frac{x^2(x^2)}{x^4+(x^2)^2} =\frac{x^4}{2x^4} =\frac{1}{2}.
Since approaching (0,0) along different paths produces different values, the limit does not exist.
This example demonstrates why checking only the coordinate axes is not always sufficient. Sometimes a more creative choice of path is needed to reveal the behavior of the function.
There is no fixed recipe for choosing paths. Experience helps you recognize substitutions that simplify the expression or reveal different behavior.
Continuity in Multivariable Functions
\lim_{(x,y)\to(a,b)} f(x,y)=f(a,b)
A function is continuous at a point if its limit at that point exists and equals the function’s value. A function is continuous in a region if it is continuous at every point in that region. Polynomial functions are continuous everywhere. Rational functions are continuous wherever their denominator is nonzero. Compositions of continuous functions are also continuous, provided each function is defined.
Establishing continuity is important because it allows us to evaluate limits by direct substitution.
For example, let
f(x,y)=x^2+3xy-y^2
Since f(x,y) is a polynomial, it is continuous everywhere. Therefore,
\lim_{(x,y)\to(1,2)}(x^2+3xy-y^2) =1^2+3(1)(2)-2^2 =3
Reflections
So far, nothing is particularly challenging here. It’s a logical continuation of run-of-the-mill Calculus. Carry on…