Are there alternative methods for proving multivariable limits?

[biggins]

For multivariable limits, the way my math books has taught me to prove they exist is to use the epsilon delta argument (for every epsilon > 0, there is a delta >0 ...). I have heard that for most cases you will almost never have to use this argument. Is this true? I know you can use the squeeze theorem on some cases but what about the others?

-hamilton
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[HallsofIvy]

I doubt that your books taught you that as "the" way to prove limits exist. That is, of course, the definition and so is always introduced first. But all of the "properties" of limits that are true for single variable limits are true for multivariable limits. For example, if your function f(x,y,z) is a fraction U(x,y,z)/V(x,y,z), the limit as (x,y,z) goes to (a,b,c) of U if L and the limit as (x,y,z) goes to (a,b,c) is M, not equal to 0, then the limit of f as (x,y,z) goes to (a,b,c) is L/M.

 

[tacman]

The epsilon-delta argument is a common method used to prove the existence of multivariable limits, but it is not the only method. In fact, for many cases, it may not be necessary to use this argument at all. The squeeze theorem, as you mentioned, can also be used to prove the existence of multivariable limits, as well as other techniques such as L'Hopital's rule or direct substitution.

It is important to understand the concept of multivariable limits and the different methods that can be used to prove their existence. While the epsilon-delta argument may be the most commonly taught method, it is not always the most efficient or necessary approach. It is always a good idea to explore different techniques and choose the one that best fits the problem at hand.

In summary, while the epsilon-delta argument is a useful tool for proving the existence of multivariable limits, it is not always necessary to use it. Other methods such as the squeeze theorem and direct substitution can also be effective in certain cases. It is important to have a good understanding of all these techniques in order to approach multivariable limits problems effectively.