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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
MCQ
If function $f(x)=\left\{\begin{array}{ll}x-\frac{|x|}{x}, & x<0 \\ x+\frac{|x|}{x}, & x>0 \\ 1, & x=0\end{array}\right.$, then
  • A
    $\lim _{x \rightarrow 0^{-}} f(x)$ does not exist
  • B
    $\lim _{x \rightarrow 0^{+}} f(x)$ does not exist
  • ✓
    $f(x)$ is continuous at $x=0$
  • D
    $\lim _{x \rightarrow 0^{-}} f(x) \neq \lim _{x \rightarrow 0^{+}} f(x)$
✓

Answer

Correct option: C.
$f(x)$ is continuous at $x=0$
(c) : We have, $f(0)=1$
$
\begin{aligned}
& \begin{aligned}
& \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0}\left(x-\frac{|x|}{x}\right)=\lim _{x \rightarrow 0}\left(x+\frac{x}{x}\right) \\
&=\lim _{x \rightarrow 0}(x+1)=1 \\
& \lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0}\left(x+\frac{|x|}{x}\right)=\lim _{x \rightarrow 0}\left(x+\frac{x}{x}\right) \\
&=\lim _{x \rightarrow 0}(x+1)=1
\end{aligned}
\end{aligned}
$
Thus, $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)$
$\therefore f(x)$ is continuous at $x=0$.

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