For example,
$\displaystyle \lim_{(x,y,z)\rightarrow(0,0,0)} \frac{x^2y^2z^2}{x^2+y^2+z^2}$
This question is from 8th edition of Stewart Calculus textbook. My fellow graduate student TAs and myself recently got many students ask this question in office hour/by email.
Because this is a freshman level calculus class, most instructors choose to only briefly explain this topic and (probably) do not expect students to write a full proof of such a problem on the exams. In our single variable calculus course syllabus the $\epsilon-\delta$ definition of limit is also omitted.
My question is, for a few people who do want to know how to solve this problem, what's the best way to explain?
Some thought
- Convince them they must know the rigorous limit definition for such a problem. (Any better approach?)
- Bound the function by 0 and $(x^2+y^2+z^2)^2$ and apply squeeze theorem. (Almost as hard as 1.?)
- Change to spherical coordinate. (They won't learn this until they get to the multivariable integration chapter.)