Question

Lim -> 0,0 of xy⁴/(x⁴+y⁴)

a) Converges to 0
b) Converges to 1
c) Diverges
d) Converges to [infinity]

Answer 1

The limit of xy^4/(x^4+y^4) as (x, y) approaches (0, 0) converges to 0.

Step-by-step explanation:

The limit as (x, y) approaches (0, 0) of the function xy4/(x4+y4) is being evaluated to determine its behavior near the origin. To find this, one method is to analyze a path approach to the limit. For instance, if we let y = mx where m is some constant, then we would substitute mx for y into the equation to see if the limit depends on the path taken. However, in this case, the power of x in the numerator is less than the power of x in the denominator, so as x approaches 0, the function will approach 0 regardless of the value of m. Therefore, the limit converges to 0.