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Calculus Level 5

$\lim_{x\to0} \dfrac{\sinh(\sin x) - \sin (\sinh x) }{x^A}$

For some constant $A$ , the limit above is finite and non-zero. Let the $L$ denote the value of this limit. Find $A\div L$ .

The answer is 315.

1 solution

Otto Bretscher
Dec 19, 2015

$\sinh\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+..\right)-\sin\left(x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+..\right)=x-\frac{x^5}{15}+\frac{x^7}{90}-..-\left(x-\frac{x^5}{15}-\frac{x^7}{90}\right)=\frac{x^7}{45}+..$ Thus $A=7, L=\frac{1}{45}$ and $\frac{A}{L}=7*45=\boxed{315}$

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The answer is 315.

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