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  • Provide Solutio for RD Sharma Class 12 Chapter Continuity Exercise 8.1 Subquestion (v) Question 10

Provide Solutio for RD Sharma Class 12 Chapter Continuity Exercise 8.1 Subquestion (v) Question 10

Answers (1)

Answer:

                Discontinuous

Hint:

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

Solution:

Given,

Given,

f(x)=\left(\begin{array}{c} \frac{1-x^{n}}{1-x}, \text { if } x \neq 1 \\\\ n-1, \text { if } x=1 \end{array}\right)

Here,  f\left ( 1 \right )=n-1

\begin{aligned} &\lim _{x \rightarrow 1} f(x)=\lim _{x \rightarrow 1} \frac{1-x^{n}}{1-x} \\\\ &\lim _{x \rightarrow 1} f(x)=\lim _{x \rightarrow 1}\left[(1-x)^{n-1}+n C_{1}(1-x)^{n-2} x+n C_{2}(1-x)^{n-3} x^{2}+\ldots .+n C_{n-1}(1-x)^{0} x^{n-1}\right] \end{aligned}  

                                                                                               \left[\because(a+b)^{n}={ }^{n} C_{0} a^{n} b^{0}+{ }^{n} C_{1} a^{n-1} b^{1}+{ }^{n} C_{2} d^{n-2} b^{2}+\ldots+{ }^{n} C_{n} a^{0} b^{n}\right]

\lim _{x \rightarrow 1} f(x)=0+0+\ldots+(1)^{n-1}=1 \neq f(1)

Thus, f\left ( x \right ) is discontinuous at  x=1.

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