Question

Let $f\left(x\right)=\{\begin{array}{cc}cos\left[x\right],& x\ge 0\\ |x|+a,& x<0\end{array}$. Then find the value of $a$, so that $\underset{x\to 0}{lim}$ f(x) exists, where [x] denotes the greatest integer function less than or equal to x.

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is B 1

Given $\underset{x\to 0}{lim}f\left(x\right)$ exists.
$\Rightarrow \underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}f\left(x\right)$
LHL $=\underset{h\to 0}{lim}f(0-h)$
$=\underset{h\to 0}{lim}|0-h|+a=a$
RHL $=\underset{h\to 0}{lim}f(0+h)$

$=\underset{h\to 0}{lim}cos[0+h]=1$
$\therefore a=1$