Q89 / 180 · COMEDK 2014 · Mathematics

If $f(x)=\left\{\begin{array}{cc}{[x]+[-x],} & x \neq 2 \\ K, & x=2\end{array}\right.$, then $f(x)$ is continuous at $x=2$, provided $K$ is equal to

  • A
    2
  • B
    $1 \quad$
  • Correct answer
    C
    $-1 \quad$
  • D
    0

Solution

$f(x)=\left\{\begin{array}{cc}{[x]+[-x],} & x \neq 2 \\ K, & x=2\end{array}\right.$ Since, $f(x)$ is continuous at $x=2$ $\therefore \lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)=f(2)$ Now, $\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2}[x]+[-x]$ $=1+(-2)=-1$ $\quad \lim _{x \rightarrow 2^{+}} f(x)=\lim _{x \rightarrow 2^{+}}[x]+[-x]=2-3=-1$ $\Rightarrow \quad K=-1$

Hence, the correct answer is option: 'C'.

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