Let $f(x) = \lim_{y \rightarrow 0}\mspace{2mu}\frac{(1 - cos(xy))tan(xy)}{y^{3}}$. Then the number o

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Let $f(x) = \lim_{y \rightarrow 0}\mspace{2mu}\frac{(1 - cos(xy))tan(xy)}{y^{3}}$. Then the number of solutions of the equation $f(x) = sinx$, $x \in R$ is :

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Sol. [Key Concept: The limit gives $f(x) = \frac{x^{3}}{2}$; count the intersections of $y = \frac{x^{3}}{2}$ with $y = sin\ x$.]$$f(x) = \lim_{y ightarrow 0}\mspace{2mu}\frac{1 - cos(xy)}{(xy)^{2}} \cdot \frac{tanxy}{xy}\frac{(xy)^{3}}{y^{3}}$$$$\begin{matrix} & f(x) = \frac{x^{3}}{2} \ & f(x) = sinx \ & \frac{x^{3}}{2} = sinx \end{matrix}$$3 points of intersection

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