Continuity and Differentiability — Maths STD 12 Science — Question

f(-3) = |-3| + 3 = 3 + 3 = 6

$\mathop {\lim }\limits_{x \to {-3^ + }} f(x)$

$ = \mathop {\lim }\limits_{x \to {-3^ + }} - 2x$

$ = \mathop {\lim }\limits_{h \to 0} - 2( - 3 + h)$ = 6

$\mathop {\lim }\limits_{x \to - {3^ - }} f(x)= \mathop {\lim }\limits_{x \to - 3^-} |x+3| = 6$

Hence continuous at x = -3

At x = 3,

$f(3) = 6 \times 3 + 2 = 20$

$ = \mathop {\lim }\limits_{x \to {3^ - }} f(x) = \mathop {\lim }\limits_{x \to {3^ - }} ( - 2x)$

$ = \mathop {\lim }\limits_{h \to 0} - 2(3 - h)$ = - 6

$ = \mathop {\lim }\limits_{x \to {3^ + }} f(x) = \mathop {\lim }\limits_{x \to {3^ + }} (6x + 2)$

$ = \mathop {\lim }\limits_{h \to 0} [6(3 + h) + 2]$ = 20

$ \Rightarrow \mathop {\lim }\limits_{x \to - {3^ - }} f(x) \ne \mathop {\lim }\limits_{x \to {3^ + }} f(x)$

Hence f(x) is not continuous at x = 3