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I've been given a couple of multivariable limit questions to solve but I don't fully understand if L'Hopital's rule can apply. Some other people I have talked to say we can split up the limits and calculate them separately e.g.

$\lim\limits_{(x,y) \to (0,0)} \frac{x^2y + sin(y)}{y} = \lim\limits_{(x) \to (0)}x^2 + \lim\limits_{(y) \to (0)} \frac{sin(y)}{y}$

Then applying L'Hopital's Rule to get the limit to be 1, however, some other people are saying we can't use L'Hopital's Rule on multivariable limits. My understanding is that we have now separated this limit into two single variable limits so we should be able to use L'Hopital's Rule.

My second question about using L'Hopital's Rule is, if we are trying to prove that a limit doesn't exist by for example making $y = \alpha x$. If we replace all instances of $y$ with $\alpha x$, can we now use L'Hopital's Rule to simplify down the limit and show that if we have different values of $\alpha$ we would get a different limit hence the limit doesn't exist?

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When you convert a multivariable limit to a single-variable limit all the single-variable limit rules apply. This means that if you have:

$\lim\limits_{(x,y) \to (0,0)} \frac{x^2y + sin(y)}{y} = \lim\limits_{(x) \to (0)}x^2 + \lim\limits_{(y) \to (0)} \frac{sin(y)}{y}$

You can use any of the rules you would a single-variable limit. This means L'Hopital's Rule is allowed. This is also true after you have substituted an equation e.g. $y = \alpha x$.

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