Question

The value of $\underset{y\to x}{lim}\frac{{\cos }^{2}y-{\cos }^{2}x}{y^2-x^2}$

Answer 1

Now, $\underset{y\to x}{lim}\frac{({\cos }^{2}y-{\cos }^{2}x)}{(y^2-x^2)}\left(0/0\right)$ form

Now, applying L' Hospital's rule we get,

$=\underset{y\to x}{lim}\frac{-2\cos y\cdot \sin y}{2y}$

$=\underset{y\to x}{lim}-\frac{\sin 2y}{2y}=-\frac{\left(\sin 2x\right)}{\left(2x\right)}$.