Show that function f(x)= ll 3-x, & if x<1 2, & if x=1 1+x, & if x… - Vidyadip

Question

Show that function$
f(x)=\left\{\begin{array}{ll}
3-x, & \text { if } x<1 \\
2, & \text { if } x=1 \\
1+x, & \text { if } x>1
\end{array}\right.
$
is continuous at $x=1$.

✓

Answer

value of $f(x)$ at $x=1$
$
f(1)=2
$
value of (LHL)
$
\begin{aligned}
\lim _{h \rightarrow 0} f(1-h) & =\lim _{h \rightarrow 0} 3-(1-h) \\
& =2

\end{aligned}
$
value of RHL
$
\begin{aligned}
\lim _{h \rightarrow 0} f(1+h) & =\lim _{h \rightarrow 0} 1+(1+h) \\
& =2
\end{aligned}
$
hence value of function $=$ L.H.L. $=$ R.H.L.
hence function is continuous at $x=1$

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