If the function f(x)= 3x+ x- 3xx^3,x R, is continuous at x=0, then f(0) is equal to :

Question

If the function $f(x)=\frac{\sin 3x+\alpha \sin x-\beta \cos 3x}{{x}^{3}},x\in R$, is continuous at $x=0$, then $f(0)$ is equal to :

Accepted Answer

$\begin{gathered} \underset{x	o 0}{lim}f(x)=f(0)\, x=0) \ \underset{x	o 0}{lim}\frac{\sin 3x+\alpha \sin x-\beta \cos 3x}{{x}^{3}} \end{gathered}$

For limit to exist $\beta =0$

$\begin{gathered} \Rightarrow \underset{x	o 0}{lim}\frac{\sin 3x+\alpha \sin x}{{x}^{3}} \ \Rightarrow \underset{x	o 0}{lim}\frac{(3+\alpha )\sin x-4{\sin }^{3}x}{{x}^{3}} \end{gathered}$

For limit to exist $\alpha +3=0\Rightarrow \alpha =-3$

$\Rightarrow \underset{x	o 0}{lim}\frac{-4{\sin }^{3}x}{{x}^{3}}=-4=f(0)$

Solutions · 1