\( \lim _{z \rightarrow \infty} \frac{\int_{1 / 2}^{z}\left[\cot ^{-1} x\right] d x}{\int_{1 / 2...

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\( \lim _{z \rightarrow \infty} \frac{\int_{1 / 2}^{z}\left[\cot ^{-1} x\right] d x}{\int_{1 / 2}^{z}\left[1+\frac{1}{x}\right] d x} \), where [.] denotes the greatest integer function, equals
(a) 0
(b) 1
(c) \( \cot 1 \)
(d) not defined
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