 To do calculus with complex variables, we need to introduce the limit concept. We claim the limit as z approaches z0 of f of z for complex values acts like the limit for real values. Now if I just leave that statement there, I'll probably get my mathematician card for a vogue, so I'll say we'll introduce a formal approach later. In the meantime, it's useful to remember that if f is an algebraic or transcendental function defined at x equal to a, then the limit as x approaches a of f of x is f of a. So if we want to find the limit as z approaches i of z cubed, then if we pretend that this is a real limit, since z cubed is an algebraic function and defined at z equal to i, the limit is i cubed, which is negative i. It's important to understand that theorem only holds for algebraic or transcendental functions of z. For example, the argument of z can be expressed as the arctangent of y divided by x, and while arctangent is a transcendental function, it's not a transcendental function of z, so the theorem doesn't help us find the limit. Instead, we'll need to consider the real and complex parts of the function. So suppose we have our function expressed in terms of its real and complex components. So first, if the limit as z approaches z0 of f of z is some complex number, then it's necessary that the limit of the functions of two variables equal the real and complex parts. Conversely, if the limit of the real and complex parts exist, the limit of the function will be the sum of those limits. Well, don't take my word for it. You should prove these things. This seems to give us a way forward. Unfortunately, when evaluating the limits of multi-variable functions, it's important to keep in mind the limit must be the same regardless of how you approach the point. And in general, we can't assume that we could interchange the limits. And what this means is that in general, it's very difficult to prove that the limit of a complex function is equal to something. More often, we can show that the limit does not exist. Suppose we find f of z goes to some value l when z goes to z0 along some path c. In order for the limit to be l, f of z has to go to l when z goes to z0 along every other path c prime. If it isn't, the limit does not exist. So if possible, find the limit as z goes to 0 of arg z. So suppose we approach z equals 0 along the path where y is equal to x. Then along this path, arg z is always pi fourths, and so I could say arg z tends to pi fourths. But now let's try a different path. Suppose we approach z equals 0 along the path c prime where y is equal to 0. Then every point along this path has an argument of 0, and so arg z tends to 0. And since the two paths give us different limits, the limit does not exist.