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

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 $x=0$.