How do you compute the value of a multivariable limit?

Question

I've been wondering this as I've tried to solve problems by parameterizing variables to f.e x = $\lambda(t)t$ and y = $\mu(t)t$ and reducing by dividing common t factors. I've looked online for methods to compute multivariable limits and I have found no one who shows how to do it and almost all of them just "assume" a limit and use an epsilon-delta proof. Also some use x = $rcos(\theta)$ and y = $rsin(\theta)$, but since $sin^2+cos^2 = 1$ doesn't this not work?

As an example take $\lim_{(x,y)\to(0,0)}\frac{x^3y}{x^6+y^2}$. I then find $\frac{t^4\lambda^3(t)\mu(t)}{t^2(\mu^2(t)+t^4\lambda^6(t))}$ which reduces to $\frac{t^2\lambda^3(t)\mu(t)}{\mu^2(t)+t^4\lambda^6(t)}$ and thus the limit is $\frac{0}{\mu^2(t)+0} = 0$ as t approaches $0$. This limit doesn't exist however as it is path dependent.

Thus my question is: Is there a way to compute multivariable limits without "guessing" them beforehand, and if so how is it done and what is/isn't a correct method?

Thank you in advance!

Answer 1

I don't think there's a particularly generic approach that will always work. There are a good number of tricks. I'll discuss the $(x, y) \to (0, 0)$ case for simplicity.

1. Fix $m$, substitute $y=mx$, and let $x \to 0$. If the result depends on $m$, the limit does not exist. This is often a quick-and-dirty way to see if a limit "obviously" doesn't exist, but it cannot prove the limit does exist.

   Example: $\lim_{(x, y) \to 0} \frac{xy}{x^2 + y^2}$ does not exist since $\lim_{x \to 0} \frac{x(mx)}{x^2 + (mx)^2} = \frac{m}{1+m^2}$ depends on $m$.

2. Polar: substitute $x=r\cos\theta, y=r\sin\theta$ and let $r \to 0$. If the result is the same no matter what $\theta$ is doing, the limit exists and is that value. Otherwise the limit does not exist. This approach will often give insight even if it doesn't immediately work out.

   Example: $\lim_{(x, y) \to 0} \frac{x^2 y}{x^2 + y^2} = 0$ since $\lim_{r \to 0} \frac{r^3\cos^2\theta\sin\theta}{r^2\cos^2\theta+r^2\sin^2\theta} = lim_{r \to 0} r (\cos^2\theta\sin\theta) = 0$.

3. Series expansions. L'Hopital's Rule is fundamentally an example of this philosophy, so I won't write out an example. While L'Hopital's Rule is a single-variable phenomenon, you could still try a multivariable Taylor expansion. I haven't personally seen that in practice.

4. Inspired substitutions. This requires "insight", whatever that means. Often the idea is something like, "well, if $x$ were really really small compared to $y$, but not too small, then this term would dominate, and the quotient would be basically 1, so..."

   Example: $\lim_{(x, y) \to 0} \frac{x^3 y}{x^6 + y^2}$ does not exist since when $y=mx^3$, we have $\lim_{x \to 0} \frac{x^3 (mx^3)}{x^6 + (mx^3)^2} = \lim_{x \to 0} \frac{m}{1+m^2}$ depends on $m$.

Answer 2

I don’t believe there is a rule that fits all multi variable limits. Generally, one looks for continuity at the point and then failing that will apply various different techniques until something sticks (such as polar coordinates, trying many different paths then generalising, and Taylor series).

This is a good thread detailing several ways to approach such problems, hopefully it helps a bit!

Is there a step by step checklist to check if a multivariable limit exists and find its value?

If you’re looking to figure out what the limit is to aid proving it, I would try several paths and if all paths tend to the same point then I would assume that to be the result I try to prove. 3D graphing calculators can help with this!!