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**chisigma** a) if $\displaystyle (x,y) \rightarrow 0$ along a straight line, then is $\displaystyle y=k\ x$. But if $\displaystyle k > 0$ and $\displaystyle x>0$ [$\displaystyle k<0$ and $\displaystyle x<0$ leads to similar conclusions...] is $\displaystyle \lim_{x \rightarrow 0} \frac{x^{2}}{k\ x} = 0 $ and in 'proximity' of $\displaystyle x=0$ is $\displaystyle k\ x> x^{2}$ so that the limit is 0. If $\displaystyle k =0$ and $\displaystyle x \ne 0$ is $\displaystyle x^{4} > y$ and the limit is again 0...

b) as seen in a) there is almost one 'trajectory' for which the limit is 0. Now we consider the 'trajectory' defined by $\displaystyle y= |x^{3}|$ we find that the limit is 1, so that no limit exists...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$