MHB What does this Limit mean geometrically?

[evinda]

Hello! (Wave)

I want to find the following limit, if it exists.

$\lim_{(x,y) \to (0,0)} \frac{\cos x-1-\frac{x^2}{2}}{x^4+y^4}$ If we say : let $(x,y) \to (0,0)$ along the line $y=0$ , what exactly does it mean geometrically?

Also, if we want to check whether the limit $\lim_{(x,y) \to (0,0)} g(x,y) $ exists where $g(x,y)=\frac{xy}{x^2+y^2}$, we pick $y=mx$ and since $g(x,mx)$ depends on $m$ we deduce that the limit does not exist.

If $g(x,mx)$ would be independent on $m$, would that mean that the limit exists? If so, why?

 

[Prove It]

Hello! (Wave)

I want to find the following limit, if it exists.

$\lim_{(x,y) \to (0,0)} \frac{\cos x-1-\frac{x^2}{2}}{x^4+y^4}$ If we say : let $(x,y) \to (0,0)$ along the line $y=0$ , what exactly does it mean geometrically?

Also, if we want to check whether the limit $\lim_{(x,y) \to (0,0)} g(x,y) $ exists where $g(x,y)=\frac{xy}{x^2+y^2}$, we pick $y=mx$ and since $g(x,mx)$ depends on $m$ we deduce that the limit does not exist.

If $g(x,mx)$ would be independent on $m$, would that mean that the limit exists? If so, why?

What do you mean by "independent on m"?

 

[evinda]

What do you mean by "independent on m"?

I mean that the result of the limit isn't a function of $m$...

 

[evinda]

Also, if we want to check if the limit $\lim_{(x,y) \to (0,0)} \frac{\sin{(2x)}-2x+y}{x^3+y}$ exists, we can consider the limit along the line $y=0$ and the limit along $x=0$ and we will see that they are not equal. Is this the only way to see that the limit does not exist?