Simplify (cos x - cos^3 x)/x^2 and evaluate its limit as x approaches 0.

Step-by-step solution

1. 1

   Check the form. At $x=0$ the numerator is $1-1=0$ and the denominator is $0$, so the quotient is $\tfrac00$ — indeterminate, and some simplification is required before the limit can be read off.

2. 2

   Factor the numerator. Both terms share a factor of $\cos x$:

   $\cos x-\cos^{3}x=\cos x\left(1-\cos^{2}x\right).$

   Pulling out the common factor is what exposes the Pythagorean identity hiding in the bracket.

3. 3

   Apply the Pythagorean identity. Since $\sin^{2}x+\cos^{2}x=1$ gives $1-\cos^{2}x=\sin^{2}x$,

   $\cos x-\cos^{3}x=\cos x\,\sin^{2}x,$

   so the whole expression becomes $\dfrac{\cos x\sin^{2}x}{x^{2}}$. The $x^{2}$ in the denominator now has an obvious partner: $\sin^{2}x$.

4. 4

   Split into standard pieces. Group the $\sin^{2}x$ with the $x^{2}$:

   $\frac{\cos x-\cos^{3}x}{x^{2}}=\cos x\left(\frac{\sin x}{x}\right)^{2}.$

   This is the simplified form, valid for all $x\ne 0$, and it is built entirely out of quantities whose limits at $0$ are known.

5. 5

   Take the limit. Using $\lim_{x\to0}\cos x=1$ and the fundamental limit $\lim_{x\to0}\frac{\sin x}{x}=1$, together with the product and power rules:

   $\lim_{x\to0}\frac{\cos x-\cos^{3}x}{x^{2}}=1\cdot 1^{2}=1.$

   Numerical confirmation: at $x=0.1$ the quotient is $0.99169$, at $x=0.01$ it is $0.999917$, at $x=0.001$ it is $0.99999917$ ✓ — converging to $1$ from below, as $\cos x<1$ predicts.