Rewriting an expression in order to avoid cancellation
$\frac{1 - (\cos (x))^{3}}{x^{2}} .$

Question

Rewriting an expression in order to avoid cancellation            1      −      (      cos      ⁡      (      x      )              )        3                    x      2        .

Xzavier Shelton · Accepted Answer

1      −              cos        3            ⁡      x              x      2        =            2              sin        2            ⁡              x        2                    x      2        ⋅      (    1    +    cos    ⁡    x    +          cos      2        ⁡    x    )    =      1    2    ⋅            (                        sin          ⁡          (          x                      /                    2          )                          x                      /                    2                    )        2    ⋅      (    1    +    cos    ⁡    x    +          cos      2        ⁡    x    )  and now no factor is troublesome in a neighborhood of zero, since:            sin      ⁡      z        z    =      ∑          n      ≥      0                  (      −      1              )        n                    z                  2          n                            (      2      n      +      1      )      !      is an entire function.

Leonel Contreras · Accepted Answer

Why not simply extending continuously the function at x=0 ? Thanks to l'Hospital's rule, we have

lim_{x \to 0} \frac{1 - (\cos (x))^{3}}{x^{2}} = lim_{x \to 0} \frac{3 \cos^{2} (x) \sin (x)}{2 x} = \frac{3}{2} .

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