I have this doubt: I know that when I'm evaluating a limit in several variables I must be very careful because I must show that the limit is independent of the path along with it is [MATH](x,y) \to (x_0,y_0)[/MATH]. For example, it is
[MATH]\lim_{(x,y) \to (0,0)} \frac{e^{2x^2}-\cos(2y)}{x^2+y^2}=\lim_{(x,y) \to (0,0)} \frac{1+2x^2+o(x^2)-1+2y^2+o(y^2)}{x^2+y^2}=2[/MATH]Was evaluated using Taylor's expansion, but how do I deduce from this that I've shown that the limit is [MATH]2[/MATH] for every possible path in [MATH]\mathbb{R}^2[/MATH] that passes through [MATH](0,0)[/MATH]? Thank you.
[MATH]\lim_{(x,y) \to (0,0)} \frac{e^{2x^2}-\cos(2y)}{x^2+y^2}=\lim_{(x,y) \to (0,0)} \frac{1+2x^2+o(x^2)-1+2y^2+o(y^2)}{x^2+y^2}=2[/MATH]Was evaluated using Taylor's expansion, but how do I deduce from this that I've shown that the limit is [MATH]2[/MATH] for every possible path in [MATH]\mathbb{R}^2[/MATH] that passes through [MATH](0,0)[/MATH]? Thank you.