I've been doing some revision and I came across a question which boils down to deciding whether or not the following limit is equal to f(0,0) = 0.(adsbygoogle = window.adsbygoogle || []).push({});

The function is:

[tex]

f(x,y) = \left\{ {\begin{array}{*{20}c}

{x^y ,x > 0} \\

{0,otherwise} \\

\end{array}} \right.

[/tex]

The limit is:

[tex]

\mathop {\lim }\limits_{\left( {x,y} \right) - > \left( {0,y'} \right)} f(x,y) = L

[/tex]

where L is either undefined, or defined - I don't know yet and y'(y-prime or y-dash) is a negative number. Approaching (x,y') along any path in the half-plane x <= 0 gives a limit of zero which is fairly obvious. Just looking at the definition of the function I think that the only points which may be discontinuous are (0,y') where y is a negative number.

I can think of a fudge method to 'show' that the function diverges to infinity if I approach (0,y') from the half plane x > 0(where f(x,y) = x^y) but that obviously isn't sufficient. So I'm hoping that someone can explain to me how to take evaluate the limit(if it exists) of f(x,y) as (x,y) -> (0,y') where y' is a negative number. So for instance how would I evaluate the limit of f(x,y) as (x,y) approaches (0,-5)? I mean, from the half plane x <= 0 the limit is just zero, but what about from the half plane x > 0(where f(x,y) = x^y)? Any assistance would be good, thanks.

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# Limit check

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