Question

Find the continuity of f(x), If
$f\left(x\right)=\left\{\begin{array}{ccc}x+1& if& x\ge 1\\ x^2+1& if& x<1\end{array}\right\}$

Answer 1

Consider the given function.

$\left\{\begin{array}{ccc}x+1& if& x\ge 1\\ x^2+1& if& x<1\end{array}\right\}$

$x+1ifx\ge 1$

$x^2+1ifx<1$

At $x=1$

L.H.L

${lim}_{x\to 1^{-}}f\left(x\right)={lim}_{h\to 0}f(1-h)$

$={lim}_{h\to 0}(x^2+1)$

$={lim}_{h\to 0}({(1-h)}^2+1)$

$={lim}_{h\to 0}(1^2+h^2-2h+1)$

$=2$

R.H.L

${lim}_{x\to 1^{+}}f\left(x\right)={lim}_{h\to 0}f(1+h)$

$={lim}_{h\to 0}(x+1)$

$={lim}_{h\to 0}((1-h)+1)$

$={lim}_{h\to 0}(1-h+1)$

$=2$

$L.H.L.=R.H.L.$

Hence, this is the continues at $x=1$