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I am trying to calculate the following limit:

$$ \lim_{x \to 0} {x\over x + ix}$$

The top moves towards $0$ along the $x$-axis and the bottom moves towards zero along the diagonal (in the complex plane).

If this was all on the $x$-axis I could apply de l'Hopital's rule. But now that it's in two dimensions I don't know what I could do.

How to evaluate limit expressions in two and more dimensions?

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    $\begingroup$ Here all we need to do is cancel. $\endgroup$ – André Nicolas Sep 14 '15 at 5:09
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Notice, $$\lim_{x\to 0}\frac{x}{x+ix}$$ $$=\lim_{x\to 0}\frac{x}{x(1+i)}$$ $$=\lim_{x\to 0}\frac{1}{1+i}$$ $$=\frac{1}{1+i}$$

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    $\begingroup$ I really do not see the point of writing out the answer completely when Andre Nicolas has already commented on how to do it $\endgroup$ – Shailesh Sep 14 '15 at 5:56

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