Are there any general guidelines for proving limits of multivariable functions?

Question

Today I was trying to prove that

$$\lim_{(x, y) \to (0, 0)}\dfrac {x^2y^2}{x^2+y^2} = 0 $$

I got really lucky because the AM-GM inequality directly applies here to give us $$\dfrac {x^2y^2}{x^2+y^2} \le \dfrac {x^4 + y^4}{x^2+y^2} \le \dfrac {(x^2+y^2)^2}{x^2+y^2} = x^2+y^2$$

And thus we may choose $\delta = \sqrt \epsilon$.

However, this was really lucky to turn out so cleanly. My question is, when it isn't so clean, are there any general guidelines?

For example:

• Do you look at the $|f(x, y)- L| < \epsilon$ and try to manipulate it? What exactly is the goal when you try to manipulate this?
• Do you look the $\sqrt {(x-a)^2 + (y-b)^2} < \delta$ term and try to manipulate it? What exactly is the goal when you try to manipulate this?

Etc.

Thank you.

Answer 1

The answer in greatest generality is no. In smaller generality, we can check for certain properties (differentiability for instance) and we can perform simplifications which make things clearer/convenient. That second bit mostly comes from experience and intuition. But, in the VAST majority of cases there isn't anything nice to do. For example

$$\sqrt{\frac{(e^{x^2}-2x^3)^2+\cos(x)}{1+2x(\sin(x)-\tan(x^2)}}$$

isn't going to have any nice algebraic simplifications that are going to help.

Now, as far as $\varepsilon-\delta$ language goes. $|f(x,y)-L|<\varepsilon$ is what we would look at if we had a solid guess on the limit. In your example above it is pretty clear that its $0$ and so we would perform manipulations on the inequality above to figure out $\delta$ in terms of our $\varepsilon$. We don't really look at $\sqrt{(x-a)^2+(y-b)^2}<\delta$ until we have figured out the limit and worked with $|f(x,y)-L|<\varepsilon$ to figure out our $\delta$.

However, we don't really care about things in generality. We only care about certain classes of functions (linear functions for example) which have nicer properties we can exploit. In general, your intuition is your best tool. In your example, we can see the multiplication in the numerator is going to drive the numerator to $0$ much faster than the sum will in the denominator.

Answer 2

In the case of a fraction with obviously positive denominator such as $x^2 + y^2$ or $x^4 - x^2 y^2 + y^4,$ or $x^2 + y^4,$ i have grown fond of simply finding the maximum (absolute value) of the numerator with constant denominator as the constraint using Lagrange Multipliers. This is not usually quick, but leads to a level of certainty about the result.