# Evaluating limits in dimensions greater than $1$

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?

• Here all we need to do is cancel. – André Nicolas Sep 14 '15 at 5:09

Notice, $$\lim_{x\to 0}\frac{x}{x+ix}$$ $$=\lim_{x\to 0}\frac{x}{x(1+i)}$$ $$=\lim_{x\to 0}\frac{1}{1+i}$$ $$=\frac{1}{1+i}$$