Question

Consider The Function F(X,Y)=X2+Y2 / x3y.

(a) Explain why f(x,y) is continuous for all (x,y)=(0,0) and use this to compute lim(x,y)→(2,3) f(x,y).
(b) Does lim(x,y)→(0,0) f(x,y) exist?

Answer 1

(a) The function \(f(x, y)\) is continuous for all (x, y) = (0, 0), and the limit as (x, y) approaches (2, 3) is
`\((13)/(54)\)`.

(b) The limit
`\(\lim_{{(x, y) \to (0, 0)}} f(x, y)\)` does not exist.

Step-by-step explanation:

(a) The function
`\(f(x, y) = (x^2 + y^2)/(x^3y)\)` is continuous at (0, 0) if the limit of \(f(x, y)\) as \((x, y)\) approaches (0, 0) exists and is equal to f(0, 0). In this case, as \(x\) and \(y\) approach (0), the expression
`((x^2 + y^2)/(x^3y)\))` simplifies to \(\frac{1}{y}\), and as (y) approaches (0), the limit is infinite. Thus, f(x, y) is continuous at (0, 0).

To compute
`\(\lim_{{(x, y) \to (2, 3)}} f(x, y)`, substitute (x = 2) and (y = 3) into the function, resulting in
`\((13)/(54)\)`.

(b) For
`\(\lim_{{(x, y) \to (0, 0)}} f(x, y)\)` to exist, the limit must be the same along any path to (0, 0). However, if we approach (0, 0) along the (x)-axis (y = 0), the limit is (0), but if we approach along the (y)-axis (x = 0), the limit is infinite. Therefore, the limit does not exist.

Understanding limits and continuity in multivariable calculus helps analyze the behavior of functions at specific points and provides insights into their overall behavior. This is crucial for various applications, especially in physics, engineering, and optimization problems.