Given
When x = -3
If f(x) is continuous for x = -3 then
$\underset{\mathit{x}\to 3^{-}}{\lim }\mathit{f}\left(\mathit{x}\right)=\underset{\mathit{x}\to -3^{+}}{\lim }\mathit{f}\left(\mathit{x}\right)=\mathit{f}\left(-3\right)$
Finding L.H.L.
$\underset{\mathit{x}\to 3^{-}}{\lim }|\mathit{x}|+3$
$=\underset{\mathit{h}\to 0}{\lim }|-3-\mathit{h}|+3$
Putting h = 0 then we get,
= | -3 - 0 | + 3
= | -3 | + 3
= 6
Finding R.H.L.
$\underset{\mathit{x}\to -3^{+}}{\lim }-2\mathit{x}=\underset{\mathit{h}\to 0}{\lim }-2\left(-3+\mathit{h}\right)$
$=\underset{\mathit{h}\to 0}{\lim }6-2\mathit{h}$
Putting h = 0 then we get,
= 6 - 2 $\times$ 0
= 6
Find f(x) at x = -3
f(-3) = | -3 | + 3 = 6
Hence, $\underset{\mathit{x}\to 3^{-}}{\lim }\mathit{f}\left(\mathit{x}\right)=\underset{\mathit{x}\to -3^{+}}{\lim }\mathit{f}\left(\mathit{x}\right)=\mathit{f}\left(-3\right)$
Therefore, the function f(x) is continuous at x = -3.
When x = 3
If f(x) is continuous for x = 3 then
$\underset{\mathit{x}\to 3^{-}}{\lim }\mathit{f}\left(\mathit{x}\right)=\underset{\mathit{x}\to -3^{+}}{\lim }\mathit{f}\left(\mathit{x}\right)=\mathit{f}\left(3\right)$
Finding L.H.L.
$\underset{\mathit{x}\to 3^{-}}{\lim }-2\mathit{x}=\underset{\mathit{h}\to 0}{\lim }-2\left(-3-\mathit{h}\right)$
$=\underset{\mathit{h}\to 0}{\lim }-6+2\mathit{h}$
Putting h=0 then we get,
= -6 + 2 $\times$ 0
= -6
Finding R.H.L.
$\underset{\mathit{x}\to -3^{+}}{\lim }6\mathit{x}+2=\underset{\mathit{h}\to 0}{\lim }6\left(3+\mathit{h}\right)+2$
$=\underset{\mathit{h}\to 0}{\lim }18+6\mathit{h}+2$
Putting h = 0 then we get,
= 18 + 6 $\times$ 0 + 2
= 20
Find f(x) at x = 3
f(3) = 6x + 2 at x = 3
f(3) = 6 $\times$ 3 + 2 = 20
Hence, $\underset{\mathit{x}\to 3^{-}}{\lim }\mathit{f}\left(\mathit{x}\right)\mathrm{ne}\underset{\mathit{x}\to -3^{+}}{\lim }\mathit{f}\left(\mathit{x}\right)=\mathit{f}\left(-3\right)$
Therefore, the function
f(x) is discontinuous at x = 3
When x < -3,
For x < -3, f(x) = | x | + 3
Since the function f(x) = | x | + 3 is a modulus function so it is continuous.
∴ f(x) is continuous for x < -3
When x > 3,
For x > 3, f(x) = 6x + 2
Since the function f(x) = 6x + 2 is a polynomial so it is continuous.
∴ f(x) is continuous for x > 3
When -3 < x < 3
For -3 < x < 3, f(x) = -2x
Since the function f(x) = -2x is a polynomial so it is continuous.
∴ f(x) is continuous for -3 < x < 3
∴ discontinuous at x = 3 only.