Let f(x)=_y0((1-(xy))(xy))/(y^3).

Question

Let $f(x)=\displaystyle\lim_{y	o0}\frac{(1-\cos(xy))	an(xy)}{y^3}$. Then the number of solutions of the equation $f(x)=\sin x$, $x\in\mathbb{R}$, is:

Accepted Answer

Write
$$f(x)=\lim_{y\to0}\frac{1-\cos(xy)}{(xy)^2}\cdot\frac{\tan(xy)}{xy}\cdot\frac{(xy)^3}{y^3}=\frac12\cdot1\cdot x^3=\frac{x^3}{2}$$

So the equation becomes $\dfrac{x^3}{2}=\sin x$. Since both sides are odd functions, $x=0$ is always a solution. Comparing the graphs of $y=x^3/2$ and $y=\sin x$ (a cubic growing from the origin against a bounded oscillating curve), they cross at exactly one positive point and, by symmetry, one negative point, in addition to $x=0$.

Total number of solutions $=\mathbf{3}$.

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