$\lim_{x \rightarrow \pi/2}\left[x \tan x - \left(\frac{\pi}{2}\right) \sec x\right]$ is e…

This limit involves an indeterminate form where both terms blow up as $x \to \pi/2$. The key is to rewrite everything in terms of $\sin x$ and $\cos x$, then use a Taylor expansion around $x = \pi/2$ by substituting $u = x - \pi/2$ (so $u \to 0$). This transforms the messy trigonometric behavior into a polynomial approximation you can evaluate cleanly.

This limit involves an indeterminate form where both terms blow up as $x \to \pi/2$. The key is to rewrite everything in terms of $\sin x$ and $\cos x$, then use a Taylor expansion around $x = \pi/2$ by substituting $u = x - \pi/2$ (so $u \to 0$). This transforms the messy trigonometric behavior into a polynomial approximation you can evaluate cleanly.