Find the value


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

Show that $\lim _{x \rightarrow 1} f(x)$ does not exist.



Left Hand Limit(L.H.L.):

$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{-}} 1+x^{2}$




Right Hand Limit(R.H.L.):

$\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1^{+}} 2-x$




$\lim _{x \rightarrow 1^{-}} f(x) \neq \lim _{x \rightarrow 1^{+}} f(x)$

Thus, $\lim _{x \rightarrow 1} f(x)$ does not exist.


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