Answer
$\displaystyle\lim_{\theta\rightarrow \frac{\pi}{4}} \dfrac{\tan\theta-2\sin\theta\cos\theta}{\theta-\frac{\pi}{4}}=2$
Work Step by Step
We have to estimate the limit:
$\displaystyle\lim_{\theta\rightarrow \frac{\pi}{4}} \dfrac{\tan\theta-2\sin\theta\cos\theta}{\theta-\frac{\pi}{4}}$
$\dfrac{\pi}{4}\approx 0.785398$
Compute $\dfrac{\tan\theta-2\sin\theta\cos\theta}{\theta-\frac{\pi}{4}}$ for values of $\theta$ close to $0.785398$, approaching from both sides:
$\dfrac{\tan (0.7)-2\sin (0.7)\cos (0.7)}{0.7-\frac{\pi}{4}}\approx 1.6763985$
$\dfrac{\tan (0.75)-2\sin (0.75)\cos (0.75)}{0.75-\frac{\pi}{4}}\approx 1.8616369$
$\dfrac{\tan (0.77)-2\sin (0.77)\cos (0.77)}{0.77-\frac{\pi}{4}}\approx 1.9390301$
$\dfrac{\tan (0.78)-2\sin (0.78)\cos (0.78)}{0.78-\frac{\pi}{4}}\approx 1.9784846$
$\dfrac{\tan (0.785)-2\sin (0.785)\cos (0.785)}{0.785-\frac{\pi}{4}}\approx 1.9984078$
$\dfrac{\tan (0.785398)-2\sin (0.785398)\cos (0.785398)}{0.785398-\frac{\pi}{4}}\approx 1.9999993$
$\dfrac{\tan (0.786)-2\sin (0.786)\cos (0.786)}{0.786-\frac{\pi}{4}}\approx 2.0024083$
$\dfrac{\tan (0.79)-2\sin (0.79)\cos (0.79)}{0.79-\frac{\pi}{4}}\approx 2.0184641$
$\dfrac{\tan (0.8)-2\sin (0.8)\cos (0.8)}{0.8-\frac{\pi}{4}}\approx 2.0589844$
$\dfrac{\tan (0.85)-2\sin (0.85)\cos (0.85)}{0.85-\frac{\pi}{4}}\approx 2.2703364$
$\dfrac{\tan (0.9)-2\sin (0.9)\cos (0.9)}{0.9-\frac{\pi}{4}}\approx 2.4983071$
Therefore we got:
$\displaystyle\lim_{\theta\rightarrow \frac{\pi}{4}} \dfrac{\tan\theta-2\sin\theta\cos\theta}{\theta-\frac{\pi}{4}}=2$
