Answer
$\dfrac{2}{3}$
Work Step by Step
To find the limit, we apply the rule:
$\lim\limits_{x \to a} \dfrac{A(x)}{B(x)}=\dfrac{\lim\limits_{x \to a} A(x)}{\lim\limits_{x \to a} B(x)}$
$\lim\limits_{x\to \dfrac{\pi}{4}}\dfrac{\tan (x)}{1+\cos^2 (x)}\\=\dfrac{\lim\limits_{x\to \dfrac{\pi}{4}} \tan (x) }{\lim\limits_{x\to \dfrac{\pi}{4}} 1+\cos^2 (x)} \\=\dfrac{\tan (\dfrac{\pi}{4})}{1+\cos^2 (\dfrac{\pi}{4})} \\=\dfrac{1}{1+\dfrac{1}{2}} \\=\dfrac{2}{3}$
