Let $f(x)=\left\{\begin{array}{cl}\alpha+\frac{\sin [x]}{x} & , ~x>0 \\ 2 & , ~x=0 \\ \beta+\left[\frac{\sin x-x}{x^3}\right] & , ~x<0\end{array}\right.$, where [ ] denotes the greatest integer function. If f(x) is continuous at x = 0, then $\beta$ is equal to
Answer & explanation
Correct answer: option 2
$\lim\limits_{x→0}β+\left[\frac{\sin x-x}{x^3}\right]$
$=\lim\limits_{x→0}β+\left[\frac{\left(x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{7!}+....\right)-x}{x^3}\right]$
$=\lim\limits_{x→0}β+\left[-\left(\frac{1}{6}+\frac{x^2}{120}-\frac{x^4}{7!}+...\right)\right]=f(0)$
so $\lim\limits_{x→0}β-1=2$
so $β=3$
Also, $\frac{\sin x-x}{x^3}$ is an even function. Therefore, $-\frac{1}{6}<\frac{\sin x-x}{x^3}<0$ for all x.
Thus, $\left[\frac{\sin x-x}{x^3}\right]=-1$ for all x.
It is given that f(x) is continuous at x = 0.
∴ $\lim\limits_{x \rightarrow 0^{+}} f(x)=f(0)=\lim\limits_{x \rightarrow 0^{-}} f(x)$
$\Rightarrow \lim\limits_{x \rightarrow 0^{+}} \alpha+\frac{\sin [x]}{x}=2=\lim\limits_{x \rightarrow 0^{-}} \beta+\left[\frac{\sin x-x}{x^3}\right] $
$\Rightarrow \alpha+0=2=\beta-1 \Rightarrow \alpha=2, \beta=3 \Rightarrow \beta=\alpha+1$