Lim x->pi/2 sin(x-pi/2)/√x/2-√pi/4

Lim x->pi/2 sin(x-pi/2)/√x/2-√pi/4

$begin{aligned} &lim_{xrightarrowfrac{pi}{2}}frac{sin{left(x-frac{pi}{2}right)}}{sqrt{frac{x}{2}}-sqrt{frac{pi}{4}}}timesfrac{sqrt{frac{x}{2}}+sqrt{frac{pi}{4}}}{sqrt{frac{x}{2}}+sqrt{frac{pi}{4}}}\ =&lim_{xrightarrowfrac{pi}{2}}frac{sin{left(x-frac{pi}{2}right)}}{frac{x}{2}-frac{pi}{4}}timeslim_{xrightarrowfrac{pi}{2}}left(sqrt{frac{x}{2}}+sqrt{frac{pi}{4}}right)\ =&lim_{xrightarrowfrac{pi}{2}}frac{sin{left(x-frac{pi}{2}right)}}{frac{1}{2}left(x-frac{pi}{2}right)}cdotleft(frac{pi}{4}+frac{pi}{4}right)\ =&frac{1}{tfrac{1}{2}}cdotfrac{2pi}{4}\ =&frac{2pi}{2}\ =&pi end{aligned}$