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  • Please Solve RD Sharma Class 12 Chapter Continuity Exercise 8.1 Question 10 Subquestion (i) Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter Continuity Exercise 8.1 Question 10 Subquestion (i) Maths Textbook Solution.

Answers (1)

Answer:

                Continuous

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}{c} |x| \cos \left(\frac{1}{x}\right), \text { if } x \neq 0 \\\\ 0 \quad, \text { if } x=0 \end{array}\right)   

We observe,

\begin{aligned} &\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}|x| \cos \left(\frac{1}{x}\right) \\\\ &\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}|x| \lim _{x \rightarrow 0} \cos \left(\frac{1}{x}\right) \end{aligned}

\begin{aligned} &\lim _{x \rightarrow 0} f(x)=0 \times \lim _{x \rightarrow 0} \cos \left(\frac{1}{x}\right)=0 \\\\ &\lim _{x \rightarrow 0} f(x)=0 \end{aligned}

The limit of function at x tends to 0 is equal to the value of function at that point hence it is continuous.

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

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