Suppose
$$f(z) = \frac{1}{(z-2)^5z}$$ is given.
By looking function, i will tell there is a $5$th-order pole at $z=2$ which is in fact true.
But on the other hand suppose
$$f(z) = \frac{\sin z}{z^5}$$ is given.
By looking just function i will again tell there is a $5$th-order pole at $z=0$, beep wrong answer says book.
If we look into Laurent series we see that highest negative power of $z-z_0$ where $f(z)$ is not defined is $4$.
At the end of day, what i want to ask is, while determining order of pole, do we need to look at laurent series?
When one can say intuitively order of pole just by looking to function?
Hint: While finding the order of a pole at z=a for a rational function f(z)/g(z) , just keep in mind that at point z=a, f(z) should be analytic and non-zero there.