- #1
evinda
Gold Member
MHB
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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?
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?