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I have to find the limit of (x^3y^3-1)/(xy-1) as (x,y) approaches (1,1). The limit alone x=1, y=1 and y=mx is 3. However, I cannot conclude that th limit is 3 since I have not tried all possible paths. So, how can I be sure that I have tried all possible paths through the point (1,1)?

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    i meant the limit along x=1,y=1 and y=mx is 3 – 2017-02-27
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    You have to argue using an arbitrary sequence $(x_k,y_k)_{k\in\mathbb{N}}$ converging to $(1,1)$. – 2017-02-27
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    Please use [$\rm \LaTeX$](http://meta.math.stackexchange.com/q/5020/290189). – 2017-02-28

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$$\lim_{(x,y)\to(1,1)} \frac{x^3y^3-1}{xy-1}=\lim_{(x,y)\to(1,1)} \frac{(xy-1)(x^2y^2+xy+1)}{xy-1}=3$$

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    I see that it is much easier when I factorise. However, I feel like I still have a problem. what happens when I cannot factorise. For example, limit of (xy^2)/(x^2+y^2) as (x,y) approaches (0,0). I found that along x=0, y=0 and y=mx, the limit is 0. But I know that zero is not necessarily the answer. how do I go about this one? – 2017-02-27
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    Notice that in this case the limit exists $=\lim(x^2y^2+xy+1)$ but in this case $\dfrac{xy^2}{x^2+y^2}$ clearly $x^2+y^2$ is 0 in origin so this point makes problem. – 2017-02-27