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

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

Answers (1)

Answer:

                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.

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