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What is $$\lim_{(x,y)\to (0,0)}xy{f(x,y)}\,\,?$$

With $\,f(x,y)=g(x)h(y)\,$,, $\,g(x)\,$ and $\,h(y)\,$ are continuous functions.

Of course the answer is 0, but I hadn't seen the $\,xy\,$ in front of the limit, so I didn't understand why the limit was supposed to be $\,0\,$. This question is obsolete, sorry for posting before reading the exercise properly.

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What are $g$ and $h$? What limit are you taking? What have you tried so far? –  Clive Newstead Oct 9 '12 at 12:34
    
And also: the limit when what goes to where? –  DonAntonio Oct 9 '12 at 12:37
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closed as not a real question by martini, Thomas, tomasz, MJD, Noah Snyder Oct 10 '12 at 9:30

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

1 Answer

up vote 1 down vote accepted

Hints:

$$(1)\;\;\;\lim_{(x,y)\to (0,0)}\,xy=0$$

$$(2)\;\;\;\text{There exist neighbourhoods}\,\,I,J\subset \Bbb R\,\,\text{ of zero and constants}\,\, M_1\,,\, M_2\,\, s.t.$$

$$|f(x)|\leq M_1\,,\,|g(y)|\leq M_2\,\,,\,\,\forall\,\,x\in I\,,\,y\in J$$

And now yes: of course the limit is zero...

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