Greatest and smallest value of a function in $\mathbb{R}^2$

Question

Does the function $$ f(x,y) = 2x+2y $$ have a greatest or smallest value in $\mathbb{R}^2$? I thought that since $$ \lim_{x \to \infty} f(x,0) = \infty $$ $$\lim_{x \to -\infty} f(x,0) = -\infty $$ then the function can be infinite great and small, am I thinking right?

Answer 1

Does the function $ f(x, ~y) = 2x + 2y $ have a greatest or smallest value in $ \mathbb{R}^{2} $? I thought that since $ \lim_{x \to \infty} f(x, ~0) = \infty $ and $ \lim_{x \to -\infty} f(x, ~0) = -\infty $ then the function can be infinite great and small, am I thinking right?

This is a plane and planes have no "values" of this type! The calculation is correct but the meaning is not.