#### Please Solve RD Sharma Class 12 Chapter 8 Continuity Exercise 8.1 Question 14 Maths Textbook Solution.

$f\left ( x \right )$ is discontinuous at the point $x=0$

Hint:

For a function to be continuous at a point, its LHL RHL and value at that point should be equal.

Solution:

Given,

$f(x)=\left(\begin{array}{ll} 3 x-2, & \text { if } x \leq 0 \\\\ x+1, & \text { if } x>0 \end{array}\right)$     at  $x=0$

$f(x)=\left(\begin{array}{l} 3 x-2, \text { if } x<0 \\\\ -2, \text { if } x=0 \\\\ x+1, \quad \text { if } x>0 \end{array}\right)$

We observe

[LHL at $x=0$  ]

$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(-h)=\lim _{h \rightarrow 0} 3(-h)-2=-2$

[RHL at $x=0$ ]

$\begin{gathered} \lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} f(0+h)=\lim _{h \rightarrow 0} f(h)=\lim _{h \rightarrow 0}(h+1)=1 \\\\ \lim _{x \rightarrow 0^{+}} f(x) \neq \lim _{x \rightarrow 0^{-}} f(x) \end{gathered}$

Thus, $f\left ( x \right )$  is discontinuous at $x=0$.