If f(x)= l 1+x^2, when 0 x 1 1-x, when x>1 ., then - Vidyadip

MCQ

If $f(x)=\left\{\begin{array}{l}1+x^2, \text { when } 0 \leq x \leq 1 \\ 1-x, \text { when } x>1\end{array}\right.$, then

A
$\lim _{x \rightarrow 1^{+}} f(x) \neq 0$
B
$\lim _{x \rightarrow 1^{-}} f(x) \neq 2$
✓
$f (x)$ is discontinuous at $x=1$
D
$f (x)$ is continuous at $x=1$

✓

Answer

Correct option: C.

$f (x)$ is discontinuous at $x=1$

(C)
$\lim _{x \rightarrow 1^{+}} f (x)=\lim _{x \rightarrow 1}(1-x)=0$
$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1}\left(1+x^2\right)$
$=1+1^2$
$=2$
$\therefore \quad \lim _{x \rightarrow 1^{+}} f (x) \neq \lim _{x \rightarrow 1^{-}} f (x)$
$\therefore f (x)$ is discontinuous at $x=1$.

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