asianese
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- 2012
Want to get these out before sleepin:
Find the limit, or show it doesn't exist:
![](https://latex.codecogs.com/png.latex?\bg_white \lim_{(x,y)\to(0,0)} \dfrac{x^3+y^3}{x^2-y^2})
The usual tricks like factorising/cancelling, going polar, x=y, x=0, y=0 don't seem to work? Probs blanking out. I know it doesn't exist but I can't find a path which isn't 0.
Next:
![](https://latex.codecogs.com/png.latex?\bg_white $Suppose $ f:\mathbb{R}\to \mathbb{R} $ is a smooth function having three or more continuous derivatives. $)
![](https://latex.codecogs.com/png.latex?\bg_white $Define $ G:\mathbb{R}^2\to\mathbb{R} $ by $)
![](https://latex.codecogs.com/png.latex?\bg_white G(x,y) = \left\{\begin{array}{lr} \dfrac{f(y)-f(x)}{y-x} & : y\ne x\\ f'(x) & : y=x\end{array}\right.)
![](https://latex.codecogs.com/png.latex?\bg_white $Use any method to calculate the partial derivative $ \dfrac{\partial G}{\partial y} $ at $ (x,x). $ Hint: the recommended method is to replace $ f(y) $ by its Taylor polynomial $T_3(y)$ about $y=x. $ Methods based on l'Hopital's rule or the chain rule for partial derivatives will also work.$)
I get the partial deriv is f'(x) but i definitely did something wrong
Find the limit, or show it doesn't exist:
The usual tricks like factorising/cancelling, going polar, x=y, x=0, y=0 don't seem to work? Probs blanking out. I know it doesn't exist but I can't find a path which isn't 0.
Next:
I get the partial deriv is f'(x) but i definitely did something wrong
Last edited: