To compute limit of z log |z| as z tends to 0 rewrite z as r (cos a + i sin a), where r=|z| and a is the argument of z. The condition that z tends to 0 is equivalent to the condition that the positive real number r tends to 0 while a could vary in arbitrary fashion.

Now when r is small, the quantity r log r is also small, as can be seen by applying the usual (real) L'Hopital's rule. If you multiply this small number by the number (cos a + i sin a), the result is still a small number since (cos a + i sin a) has unit length. This shows that the answer to the original question is 0.

Now when r is small, the quantity r log r is also small, as can be seen by applying the usual (real) L'Hopital's rule. If you multiply this small number by the number (cos a + i sin a), the result is still a small number since (cos a + i sin a) has unit length. This shows that the answer to the original question is 0.