(Solved): a) [6 pts] Compute the limit: \( \lim _{x \rightarrow \infty} x^{\left[\ln \l ...

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a) [6 pts] Compute the limit: \( \lim _{x \rightarrow \infty} x^{\left[\ln \left(\frac{\pi}{2}-\tan ^{-1}(x)\right)+x\right]} \). Hints: (i) Find the limit \( \lim _{x \rightarrow \infty}\left[\ln \left(\frac{\pi}{2}-\tan ^{-1}(x)\right)+x\right] \), (ii) note \( x=-\ln \left(e^{-x}\right) \) b) \( \left[6\right. \) pts] Compute the limit: \( \lim _{x \rightarrow \infty}(\cos (1 / x))^{[\ln (x)]^{e^{-x}}} \). Hint: Find the limit \( \lim _{x \rightarrow \infty}[\ln (x)]^{e^{-x}} \).