HallsofIvy said:For problems like this, I recommend converting to polar coordinates:
[itex]a= r cos(\theta) b= r sin(\theta)[/itex] so
[tex]\frac{ab}{a^2+ b^2}= \frac{r^2 sin(\theta)cos(\theta)}{r^2}[/tex]
[tex]= cos(\theta)sin(\theta)[/tex]
The distance from (0,0) to (a,b) is just r.
( Hmm, I forsee a serious problem in proving that limit exists!)
Well, in that case, there is no "epsilon-delta" argument that the limit exists for a very good reason. The limit does not exist.precondition said:umm... again sorry but
function is (ab)/(a^2+b^2) comma shouldn't be there...
sorry
A limit in mathematics is a value that a function or sequence approaches as its input or index approaches a given point. It is denoted by the symbol "lim" and is used to describe the behavior of a function at a particular point or as its input approaches infinity or negative infinity.
A limit exists if the value of the function at the given point is equal to the value of the limit. This means that as the input approaches the given point, the function does not have any abrupt changes or discontinuities.
The expression (a,b)->(0,0) indicates that the limit is being evaluated at the point (0,0), which is known as the origin. This means that the function is being evaluated as both a and b approach 0, and the limit is being calculated at the point (0,0).
The expression (a,b)/(a^2+b^2) is the function being evaluated in the given limit. This means that the limit is being calculated as the function (a,b)/(a^2+b^2) approaches its value at the point (0,0).
No, the limit cannot exist if the function (a,b)/(a^2+b^2) is undefined. A limit can only exist if the function is defined and approaches a specific value as its input approaches a given point. If the function is undefined, it means that its value at the given point cannot be determined, and therefore the limit cannot exist.