Limit using definition in complex plane I know that $$\lim\limits_{z \to i} \frac{z^2+1}{z}=0$$
But, I couldn't prove it using the definition of limit. Could you please help me?

Let $z = x + \mathrm i y$ and consider the limits $y=1$ while $x \to 0$ and $x=0$ while $y \to 1$, separately. 
In the case $z=x+\mathrm i$, where $x \to 0$ we get
$$\lim_{z \to \mathrm i}\frac{z^2+1}{z} = \lim_{x \to 0} \left[\frac{x(x^2+2)}{x^2+1} + \left(\frac{x^2}{x^2+1}\right)\mathrm i\right] = 0+0 \mathrm i$$
In the case $z=0+\mathrm i y$, where $y \to 1$, we get
$$\lim_{z \to \mathrm i}\frac{z^2+1}{z} = \lim_{y \to 1} \left[ 0
+\left(\frac{y^2-1}{y}\right)\mathrm i \right] = 0 + 0\mathrm i$$
