Are There General Rules for Limits in Multivariable Real-Valued Functions?

[Diophantus]

When dealing with real valued functions (one output for now) of more than one real variable, can the usual rules from R --> R be generalised in the natural way? Specifically the sum, product, quotient and composite rules. Any pathological cases?

Also I was also wondering if there are any tools analogous to L'Hopital lying around anywhere.

 

[HallsofIvy]

Looks to me like you are talking about Rⁿ--> R: a real valued function of several values. Yes, the standard rules for derivatives apply to partial derivatives and the gradient vector.

 

[Diophantus]

Yes I know that, I was more concerned about actual limits at a pedantic epsilon-delta level. Eg. for the composite rule if f(x,y) = h(g(x,y)), and we know that

lim g(x,y) = L
(x,y)-->(a,b)

then can we always say that

lim f(x,y) = lim h(z)
(x,y)-->(a,b) z-->L

 

[matt grime]

Derivatives *are* pedantic epsilon and delta arguments. And they all pass through without alteration in the R^m case: just replace | | with || ||, the Euclidean distance in R^m.

It is things like the mean value theorem that fail.

 

[HallsofIvy]

Hmm, looking back at your original post I don't know where I got the idea that you were talking about derivatives!

Yes, the "composition" law applies to limits of functions of several variables.

 

[Diophantus]

Thanks guys