Finding a two-variable function that is distinct from another on every open disk, with specifics.

Question

Consider the two-variable function $$f(x, y) = \sin(x) + \cos(x) + y^2.$$

Find a two-variable function $g(x, y)$ that is distinct from $f(x, y)$ on every open disk which contains the point $(1, 2)$ but still satisfies $$\lim_{(x,y) \to (1,2)} \frac{f(x,y) - g(x,y)}{\sqrt{(x-1)^2 + (y-2)^2 }} = 0$$

Note: Take the time to review the concept of derivatives for single and multi-variable functions, and how they work as linear approximations.

I'm having trouble finding exactly what to do for this problem, or how it relates to derivatives as linear approximations. Could someone get me started in the right direction?

Answer 1

Hint: Can we get $f(x,y) - g(x,y)=(x-1)^2 + (y-2)^2?$

Answer 2

In the one dimensional case, when you take a derivative, you find the slope of the function in a certain point p, so you can take a linear approximation of the function "not so far" from that point p.

In n dimensions, what would happen if you consider the derivatives (and therefore the slopes) with respect to the other variables?