So it was my understanding that the definition of differentiability when it came to multivariable was the following: Where $f(x,y)$ is the function and $L(x,y)$ is the linear approximation. $$\lim_{(x,y) \to (0,0)} \frac{|f (x, y) − L(x, y)|} {|\sqrt{x^2 + y^2}|}= 0$$
But when I was solving problems, there was a question that asked me to define what differentiability was and so I answered using the expression above. But in the answers they used this expression. So I was wondering what the difference was and why it looks like this. By the way the question was about checking if $f$ was differentiable at $(0,0)$ so I get why the denominator looks the way it does so I just don't get why the numerator looks the way it does. $$\lim_{h_1,h_2 \to 0} \frac{f (x_0 + h_1, y_0 + h_2) − f (x_0, y_0) − L(h_1, h_2)} {\sqrt{h^2_1 + h^2_2}}= 0$$