Question

Please anyone give the answer asap...​

Answer 1

$\large\underline{\sf{Solution-}}$

Given function is

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:f(x) = \begin{cases} &\sf{x + 3, \: \: \: for \: x \leqslant - 3} \\ &\sf{2x, \: \: \: for \: - 3 < x < 3}\\ &\sf{6x + 2, \: \: \: for \: x \geqslant 3} \end{cases}\end{gathered}\end{gathered}$

Let first we check the continuity at x = - 3.

Consider, LHL at x = - 3

$\rm :\longmapsto\:\displaystyle\lim_{x \to -3^-} \: f(x)$

$\rm :\longmapsto\:\displaystyle\lim_{x \to -3^-} \: x + 3$

$\red{\rm :\longmapsto\:Put \: x = - 3 - h, \: as \: x \: \to \: - 3, \: h \: \to \: 0}$

$\rm \: = \: \: \displaystyle\lim_{h \to 0} \: ( - 3 - h + 3)$

$\rm \: = \: \: \displaystyle\lim_{h \to 0} \: ( - h)$

$\rm \: = \: \: 0$

Consider RHL at x = - 3

$\rm :\longmapsto\:\displaystyle\lim_{x \to -3^ + } \: f(x)$

$\rm \: = \: \: \displaystyle\lim_{x \to -3^+ } \: 2x$

$\red{\rm :\longmapsto\:Put \: x = - 3 + h, \: as \: x \: \to \: - 3, \: h \: \to \: 0}$

$\rm \: = \: \: \displaystyle\lim_{h \to \: 0 } \: 2( - 3 + h)$

$\rm \: = \: \: - 6$

Since,

$\rm :\longmapsto\:\displaystyle\lim_{x \to -3^-} f(x) \: \ne \: \displaystyle\lim_{x \to -3^ + } f(x)$

$\bf\implies \:f(x) \: is \: not \: continuous \: at \: x = - 3$

Now, To check continuity at x = 3

Consider, LHL at x = 3

$\rm :\longmapsto\:\displaystyle\lim_{x \to 3^-} \: f(x)$

$\rm \: = \: \: \displaystyle\lim_{x \to 3^ - } \: 2x$

$\red{\rm :\longmapsto\:Put \: x = 3 - h, \: as \: x \: \to \: 3, \: h \: \to \: 0}$

$\rm \: = \: \: \displaystyle\lim_{h \to0} \: 2(3 - h)$

$\rm \: = \: \: 6$

Consider RHL at x = 3

$\rm :\longmapsto\:\displaystyle\lim_{x \to 3^ + } \: f(x)$

$\rm \: = \: \: \displaystyle\lim_{x \to 3^ + } \: 6x + 2$

$\red{\rm :\longmapsto\:Put \: x = 3 + h, \: as \: x \: \to \: 3, \: h \: \to \: 0}$

$\rm \: = \: \: \displaystyle\lim_{h \to 0} \: 6(3 + h) + 2$

$\rm \: = \: \: 18 + 2$

$\rm \: = \: \: 20$

Since,

$\rm :\longmapsto\:\displaystyle\lim_{x \to 3^-} f(x) \: \ne \: \displaystyle\lim_{x \to 3^ + } f(x)$

$\bf\implies \:f(x) \: is \: not \: continuous \: at \: x = 3$