$$f(x, y) = \begin{cases} \frac{\sin(x^{1010}y^{1012})}{x^{2020} - y^{1010}x^{1010} + y^{2020}} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0) \end{cases}$$
Hey! I have this function and have to study the existence of partial derivatives and differential in point (0,0). Have tried multiple ways but can't figure out a way to solve.
As I know from theory , i have to study the limit in point (0,0). Tried that by assuming that x = y from the second equation and replaced in the first to get: $$\lim_{{x \to 0}} \frac{{\sin(x^{2022})}}{{x^{2020}}}$$ From this point I don't know what to do. Can someone give a hint?
Do a u-sub $$\lim_{x \to 0} \frac{\sin(x^{2022})}{x^{2022}}$$ Let $u = x^{2022}$, Note that as $x$ approaches $0$, $u$ also approaches $0$ $$\lim_{u \to 0} \frac{\sin(u)}{u}$$ This limit is well known