Question

Find limit -> 0,0 f(x,y)=tan(-x²-y²)/(2x²+2y²)

a) Converges to 1
b) Converges to -1
c) Diverges
d) Converges to 0

Answer 1

The limit of f(x,y) = tan(-x²-y²)/(2x²+2y²) as (x,y) approaches (0,0) simplifies to ½(-1) giving us -½, which is not one of the provided options.

Step-by-step explanation:

The student has asked to find the limit of the function f(x,y) = tan(-x²-y²)/(2x²+2y²) as (x,y) approaches (0,0). To solve this, let's simplify the expression by factoring out 2 in the denominator and simplifying the tangent function.

The function then becomes f(x,y) = ½ tan(-x²-y²)/(x²+y²). Given that tan(-θ) = -tan(θ), the function simplifies further to f(x,y) = ½ (-tan(x²+y²)/(x²+y²)).

As (x,y) approaches (0,0), the expression x²+y² approaches 0. The function tan(θ)/θ approaches 1 as θ approaches 0. Hence, the function effectively becomes ½ (-1) which is ½. Therefore, the limit of the function f(x,y) as (x,y) approaches (0,0) is -½, which is not listed in the provided options.