<s>[INST] Help with finding limit of function with 2 variables Find $$\lim_{(x,y)\to(0,0)} \frac{e^{\sin(x^2+y^2)}-1}{x^2+y^2}$$
I have tried taking limits when $y=x$ and when $y=-x$ but failing to get close to an answer.
 [/INST] *

*Method 1. One may write, as $x^2+y^2 \to 0$, by the use of Taylor series expansions, $$ \begin{align}
   \sin(x^2+y^2)&=x^2+y^2+O((x^2+y^2)^3)
   \\\\e^{\sin(x^2+y^2)}&=1+x^2+y^2+O((x^2+y^2)^2) \end{align} $$
giving, as $x^2+y^2 \to 0$, $$
   \frac{e^{\sin(x^2+y^2)}-1}{x^2+y^2}=1+O(x^2+y^2). $$

*Method 2. One may write, as $x^2+y^2 \to 0$,    $$ \frac{e^{\sin(x^2+y^2)}-1}{x^2+y^2}=\frac{\color{red}{e^{\sin(x^2+y^2)}-1}}{\color{red}{\sin(x^2+y^2)}}\times
   \frac{\color{blue}{\sin(x^2+y^2)}}{\color{blue}{x^2+y^2}}. $$


Can you take it from here?
</s>