Analyzing the Limit of the Function ( f(x, y) ) as ( (x, y) to (0,0) )
The function ( f(x, y) frac{x^a y^b}{x^2 y^2} ) for ( x y eq 0 ) and ( f(x, y) 0 ) for ( x y 0 ) is often encountered in advanced calculus and real analysis. To determine the behavior of ( f(x, y) ) as ( (x, y) to (0,0) ), we can transform the limit into polar coordinates. This method provides a powerful way to analyze the function at the origin.
Transformation into Polar Coordinates
First, let's convert ( x ) and ( y ) into polar coordinates. In polar coordinates, ( x r cos theta ) and ( y r sin theta ). Substituting these into the function, we get:
[ f(x, y) frac{r^a cos^a theta cdot r^b sin^b theta}{r^2} r^{a b-2} cos^a theta sin^b theta ]
Taking the limit as ( (x, y) to (0,0) ) is equivalent to taking the limit as ( r to 0 ), since ( r ) represents the distance from the origin.
Case 1: ( a b > 2 )
If ( a b > 2 ), then ( a b - 2 > 0 ). As ( r to 0 ), ( r^{a b-2} to 0 ). Therefore, regardless of the values of ( theta ), the term ( cos^a theta sin^b theta ) remains bounded. By the Squeeze Theorem:
[ -r^{a b-2} leq r^{a b-2} cos^a theta sin^b theta leq r^{a b-2} ] Since both the upper and lower bounds approach 0 as ( r to 0 ), the function ( f(x, y) ) also tends to 0. Hence:
[ lim_{(x, y) to (0,0)} f(x, y) 0 ]
Case 2: ( a b 2 )
If ( a b 2 ), then ( a b - 2 0 ). The function simplifies to:
[ f(x, y) cos^a theta sin^b theta ]
Since ( cos^a theta ) and ( sin^b theta ) can vary depending on ( theta ), the limit is dependent on the value of ( theta ). For instance, if ( theta 0 ) or ( theta pi/2 ), ( cos^a theta sin^b theta 0 ), but if ( theta pi/4 ), ( cos^a theta sin^b theta ) could be non-zero. This implies that the limit does not exist without specifying the path of approach. For example:
1. If ( theta 0 ) or ( theta pi ), then ( cos^a theta 1 ) and ( sin^b theta 0 ), hence the limit is 0.
2. If ( theta pi/4 ), then ( cos^a theta sin^b theta eq 0 ), and the limit is not necessarily 0.
Thus, the limit does not exist when ( a b 2 ).
Case 3: ( a b
If ( a b
1. If ( theta 0 ) or ( theta pi ), then ( cos^a theta 1 ) and ( sin^b theta 0 ), hence the limit is 0.
2. If ( theta pi/4 ), then ( cos^a theta sin^b theta eq 0 ), and the limit could be ( pm infty ).
This again shows that the limit does not exist when ( a b
Conclusion
In summary, the behavior of the function ( f(x, y) frac{x^a y^b}{x^2 y^2} ) as ( (x, y) to (0,0) ) is summarized as follows:
If ( a b > 2 ), then the limit is 0. If ( a b 2 ), the limit does not exist. If ( a bThis analysis uses polar coordinates and the Squeeze Theorem to determine the limits, providing a comprehensive understanding of the function's behavior near the origin.