Order of a $0$ of a rational function

Question

I want to find the order of the zero $z = 0$ of the function $f(z) = \frac{6\sin z - 6z +z^3}{z^2}$.

I don't really know how to solve these problems except by appealing to the definition of the order of a $0$ and differentiating the function until $f^{(m)}(z_0)$ doesn't equal $0$ and then $m$ is the order.

Is there a better method for functions like this where it's hard to do that?

Answer 1

Write out the Taylor series and recognize that

$$\sin z = z - \frac 1 6 z^3 + \frac 1 {120}{z^5} + O(z^7)$$

so that

$$f(z) = \frac 1 {120} z^3 + O(z^6)$$

has a zero of order $3$ at the origin.

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On the other hand, knowing the Taylor series up to this many terms is basically the same as taking a bunch of derivatives....