#### Please Solve RD Sharma Class 12 Chapter Continuity Exercise 8.1 Question 6 Maths Textbook Solution.

$x=0$  (Discontinuous)

Hint:

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

Solution:

Given,

$f(x)=\left(\begin{array}{cc} e^{\frac{1}{x}}, \text { if } x \neq 0 \\\\ 1 & , \text { if } x=0 \end{array}\right)$

We observe,

[LHL at  $x=0$ ]

$\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(-h)=\lim _{h \rightarrow 0} e^{\frac{-1}{h}}=\lim _{h \rightarrow 0}\left(\frac{1}{e^{\frac{1}{h}}}\right)=\frac{1}{\lim _{h \rightarrow 0} e^{\frac{1}{h}}}=0$                  $\left[\because \frac{1}{\infty}=0\right]$

[RHL at $x=0$ ]

$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} f(0+h)=\lim _{h \rightarrow 0} f(h)=\lim _{h \rightarrow 0} e^{\frac{1}{h}}=\infty$

Given

$f\left ( 0 \right )=1$

It is known that for a function $f\left ( x \right )$ is to be continuous at $x=a$,

$\lim _{x \rightarrow a^{+}} f(x)=\lim _{x \rightarrow a^{-}} f(x) \neq f(a)$

But here

$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{-}} f(x) \neq f(0)$

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