Question

Let f: $R\to R$ be defined by$f\left(x\right)=\{\begin{array}{cc}\alpha +\frac{sin\left[x\right]}{2}& ifx>0\\ 2& ifx=0\\ \beta +\left[\frac{sinx-x}{x^3}\right]& ifx<0\end{array}$where $\left[y\right]$ denotes the integral part of y. If f is continuous at $x=0,then\beta -\alpha =$

• A. -1
• B. 1
• C. 0
• D. 2

Answer 1

The correct option is B

1

$x>0\Rightarrow \left[x\right]=0$
$x<0\Rightarrow \left[\frac{sinx-x}{x^3}\right]=-1$
$\therefore \propto =2=\beta -1$
$\therefore \beta -\propto =1$