X- 2 lim tan2x - x- 2

lim_x→π/2tan 2 xx π/2= Indeterminate formApplying L'Hospital Rule #160; lim_x→π/2sec^2 2x× (2)1 0=sec^2π=1 ...

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L'Hospital Rule to Remove Indeterminate Form

x→π/ 2 lim ta...

Question

$l i m x \to \frac{π}{2} \frac{t a n 2 x}{x - \frac{π}{2}}$

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l i m x \to π / 2 {tan}^{2} x [{(2 {sin}^{2} x + 3 sin x + 4)}^{1 / 2} - {({sin}^{2} x + 6 sin x + 2)}^{1 / 2}]

\frac{1}{m}

m

f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} - 2 sin x, - π \leq x \leq - \frac{π}{2} a sin x + b, - \frac{π}{2} < x < \frac{π}{2} cos x, \frac{π}{2} \leq x \leq π \end{matrix}

f (x)

[- π, π]

f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} - x - \frac{π}{2}, & x \leq - \frac{π}{2} - c o s x, & - \frac{π}{2} < x \leq 0 x - 1, & 0 < x \leq 1 l n x, & x > 1 \end{matrix}

g (x) = f (tan x) + f (cot x)

x \in (\frac{π}{2}, π)

f^{''} (x) < 0 \forall x \in (\frac{π}{2}, π),

The maximum value of $s i n (x + \frac{π}{6}) + c o s (x + \frac{π}{6})$ in the

interval $(0, \frac{π}{2})$ is attained at

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