If the function f(x)=( x)- xx^4 is continuous at each point in its domain and f(0)=1k, the…

Question

If the function $f(x)=\dfrac{\cos(\sin x)-\cos x}{x^4}$ is continuous at each point in its domain and $f(0)=\dfrac1k$, then $k$ is ..........

Answer: 6

Accepted Answer

Using Taylor expansions,
$$\cos(\sin x)=1-\frac{x^2}{2}+\frac{x^4}{6}+\frac{x^4}{24}+\cdots,\qquad\cos x=1-\frac{x^2}{2}+\frac{x^4}{24}+\cdots$$
$$\frac{\cos(\sin x)-\cos x}{x^4}\to\frac16$$
So $f(0)=\dfrac16=\dfrac1k\Rightarrow k=6$.

Solutions · 1