Question

lim as x goes to zero of: (1-2x^2-2cosx+cos^2x)/x^2

Answer 1

\[\lim_{x \rightarrow 0} \frac{ 1 - 2x^2 -2 \cos x + \cos^2 x }{ x^2 } =\]
\[\lim_{x \rightarrow 0} \frac{ 2x^2 }{ x^2 } + \lim_{x \rightarrow 0}\frac{ 1 -2cosx + \cos^2x }{ x^2 }=\]\[2 + \lim_{x \rightarrow 0} (\frac{ 1- \cos x }{ x })^2 =\]
\[2 + 0 = 2\]

Answer 2

To understand Rick's accurate answer, review lesson 5, which shows that \[\lim_{x \rightarrow 0}\frac{ \sin x }{ x}=1\] and \[\lim_{x \rightarrow 0}\frac{ \left( 1-\cos x \right) }{ x }=0\]