Why Uniformity in the Angle Matters
Polar coordinates are useful for limits at the origin because distance from the origin is represented by \(r\), while direction is represented by \(\theta\). A limit, however, concerns every point sufficiently close to the origin, not just points on a few selected rays. The angular part of an expression must therefore be controlled in a way that remains valid as the direction changes.
Write \(x=r\cos\theta\) and \(y=r\sin\theta\), where \(r>0\). To prove that a function tends to \(L\), the key goal is to bound the error from \(L\) by a quantity that depends only on \(r\) and tends to zero. Such a bound works simultaneously for every angle.
“Uniform” here means that the same bound works for all angles. It does not mean that the expression has the same value for every angle.
The Uniform-Bound Criterion
Suppose a function \(f\) is defined at all points in some punctured disk about the origin. Suppose that, in polar coordinates, there is a nonnegative function \(B(r)\) such that
for every angle \(\theta\) and every sufficiently small \(r>0\), and that \(B(r)\to 0\) as \(r\to 0^+\). Then \(f(x,y)\to L\) as \((x,y)\to(0,0)\).
This criterion follows directly from the multivariable limit definition. Near the origin, the distance from \((x,y)\) to \((0,0)\) is \(r=\sqrt{x^2+y^2}\). Given any \(\varepsilon>0\), the condition \(B(r)\to0\) provides a \(\delta>0\) such that \(0<r<\delta\) implies \(B(r)<\varepsilon\). The uniform estimate then gives an error smaller than \(\varepsilon\) at every point in that punctured disk.
Use \(x=r\cos\theta\) and \(y=r\sin\theta\) to express \(f-L\) in terms of radius and angle.
Find an inequality that is valid for every angle, not only for a chosen direction.
Verify that the resulting bound tends to zero as \(r\to0^+\).
Choose a radius small enough to make the common bound less than any prescribed \(\varepsilon>0\).
Worked Example: A Bounded Angular Factor
Worked Example: A Bounded Angular Factor
Evaluate the limit of \(f(x,y)=\dfrac{x^2y}{x^2+y^2}\) as \((x,y)\to(0,0)\), with \(f\) defined away from the origin.
For \(r>0\), substitute \(x=r\cos\theta\) and \(y=r\sin\theta\). The numerator becomes \(r^3\cos^2\theta\sin\theta\), and the denominator becomes \(r^2(\cos^2\theta+\sin^2\theta)=r^2\). Thus
Since \(|\cos\theta|\le1\) and \(|\sin\theta|\le1\), we have \(|\cos^2\theta\sin\theta|\le1\) for every \(\theta\). Therefore
The bound \(B(r)=r\) tends to zero. To check the definition explicitly, given \(\varepsilon>0\), take \(\delta=\varepsilon\). If \(0<\sqrt{x^2+y^2}<\delta\), then \(r<\varepsilon\), and the estimate gives \(|f(x,y)|\le r<\varepsilon\). Hence the limit is \(0\).
Check: The estimate was derived for every angle, and its radial bound \(r\) tends to zero. It therefore controls all points close to the origin, including points whose direction changes as they approach it.
Worked Example: A Nonzero Limit
Worked Example: A Nonzero Limit
Show that \(g(x,y)=5+\dfrac{x^2-y^2}{\sqrt{x^2+y^2}}\), defined for \((x,y)\ne(0,0)\), tends to \(5\) at the origin.
In polar coordinates, \(x^2-y^2=r^2(\cos^2\theta-\sin^2\theta)\) and \(\sqrt{x^2+y^2}=r\). Consequently,
Both \(\cos^2\theta\) and \(\sin^2\theta\) lie between \(0\) and \(1\), so their difference has absolute value at most \(1\). It follows that
Given \(\varepsilon>0\), choose \(\delta=\varepsilon\). Whenever \(0<\sqrt{x^2+y^2}<\delta\), the error satisfies \(|g(x,y)-5|\le r<\varepsilon\). Thus the limit is \(5\).
Check: The bound applies to the difference \(g-5\), not to \(g\) itself. Since that difference is bounded by \(r\), its size tends to zero while the constant part remains \(5\).
Why Fixed Rays Do Not Settle a Limit
Along a fixed ray, \(\theta\) is held constant while \(r\to0^+\). Checking every fixed ray can provide useful evidence, but it does not automatically control points that approach the origin with an angle that varies with \(r\). A proof by uniform angular bounds addresses this issue by using one estimate for all angles at each sufficiently small radius.
Worked Example: Fixed-Ray Limits but No Multivariable Limit
Worked Example: Fixed-Ray Limits but No Multivariable Limit
Consider \(h(x,y)=\dfrac{x^2y}{x^4+y^2}\), defined where the denominator is nonzero. The denominator is positive at every point other than the origin. We will show that the limit at the origin does not exist, even though the limit along each fixed ray is \(0\).
First use a ray \(y=mx\), where \(m\) is fixed. For \(m\ne0\) and \(x\ne0\), substitution gives
As \(x\to0\), the numerator \(mx\) tends to zero and the denominator tends to \(m^2>0\). Thus this ray gives limit \(0\). If \(m=0\), then \(y=0\) and \(h(x,0)=0\), so that ray also gives limit \(0\).
Now take the curved approach \(y=x^2\), with \(x\ne0\). Direct substitution yields
The values along this curve are constantly \(1/2\), whereas the fixed-ray limits are \(0\). Therefore the multivariable limit does not exist.
The polar expression reveals why the fixed-ray checks miss this approach. Substitution gives
For each fixed angle with \(\sin\theta\ne0\), the denominator tends to \(\sin^2\theta>0\) while the numerator tends to zero, so the expression tends to zero. At angles with \(\sin\theta=0\), the expression is zero. But this reasoning holds the angle fixed. The curve \(y=x^2\) approaches the positive \(x\)-axis while its angle changes; it does not stay on one fixed ray.
Check: If a uniform angular bound tending to zero existed for the absolute value of this polar expression, the value along every approach would tend to zero. The curved approach has value \(1/2\), so such a bound cannot exist.
Common Mistake: Treating a Ray Check as a Uniform Bound
A common but invalid argument is: “For every fixed \(\theta\), the expression tends to \(0\), so its limit is \(0\).” The phrase “for every fixed \(\theta\)” allows the radius required for a given error tolerance to depend on \(\theta\). A multivariable limit needs one radius that works for all directions at once.
For the last example, each fixed ray gives \(0\), but substituting \(y=x^2\) gives \(1/2\). The ray calculation therefore cannot establish the claimed limit. The correction is to seek an estimate valid for every \(\theta\), or, if no such decaying estimate is available, to investigate whether an approach with changing angle contradicts the proposed limit.
When looking for a uniform bound, useful elementary facts include \(|\sin\theta|\le1\), \(|\cos\theta|\le1\), and \(\sin^2\theta+\cos^2\theta=1\). For a product such as \(\sin^2\theta\cos^2\theta\), the identity \((\sin^2\theta-\cos^2\theta)^2\ge0\) gives \(4\sin^2\theta\cos^2\theta\le1\). Such estimates replace angle-dependent expressions by bounds that apply in every direction.
When This Method Is Useful
Uniform angular bounds are especially effective when a polar expression separates into a radial factor and an angular factor, and the angular factor remains bounded. For example, if an error has the form \(r^\alpha A(\theta)\), where \(\alpha>0\) and \(|A(\theta)|\le C\) for all \(\theta\), then
Because \(Cr^\alpha\to0\) as \(r\to0^+\), this is a uniform bound proving that the error tends to zero. The crucial condition is not simply that the radial power is positive: the angular factor must also admit a bound that does not grow uncontrollably as the angle varies.
In other problems, the expression may not separate so neatly. The same goal still guides the work: bound the entire error by a radial quantity that tends to zero. If the angle appears in a denominator, check carefully whether that denominator can become small for angles that vary with \(r\). A pointwise calculation at each fixed angle may conceal precisely that behavior.
The method gives a direct route from a polar expression to the multivariable limit definition. It also explains why testing a few directions, or even every fixed ray, is not a replacement for a bound that works throughout a punctured neighborhood.
Check Your Understanding
For each question, focus on whether the estimate applies to every angle or only to a fixed direction.
- What does it mean for a bound on a polar expression to be uniform in \(\theta\)?
- If \(|f(r\cos\theta,r\sin\theta)-L|\le 3r^2\) for all \(\theta\), what limit follows as \(r\to0^+\)?
- Why does checking the limit along each fixed ray not necessarily establish a multivariable limit?
- For an error \(r^{1/2}A(\theta)\), what additional condition on \(A\) would make a uniform-decay argument work?
- In the final worked example, what approach contradicts the fixed-ray limit, and what value does the function take along it?