All we need to do is set this up to allow us to use the fact from the notes in this section.

\[\mathop {\lim }\limits_{x \to \,0} \frac{{\cos \left( {4x} \right) - 1}}{x} = \mathop {\lim }\limits_{x \to \,0} \frac{{4\left( {\cos \left( {4x} \right) - 1} \right)}}{{4x}} = 4\mathop {\lim }\limits_{x \to \,0} \frac{{\cos \left( {4x} \right) - 1}}{{4x}} = 4\left( 0 \right) = \require{bbox} \bbox[2pt,border:1px solid black]{0}\]