Rodd and Todd

Calculus Level 5

$\lim_{x\to0} \dfrac{x - f_{2016}(x)}{x^3}$

We define a recurrence relation, $f_n(x) = \sin(f_{n-1}(x))$ for $n=2,3,4,\ldots$ , with $f_1(x) = \sin(x)$ . Evaluate the limit above.

The answer is 336.

1 solution

Otto Bretscher
Dec 19, 2015

It is not hard to see, by induction on $n$ , that the Taylor series of $f_n(x)$ is $x-\frac{n}{6}x^3+...$ . Thus the limit we seek is $\frac{n}{6}=\boxed{336}$ for $n=2016$ .

Rodd and Todd

The answer is 336.

1 solution

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