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  • Need solution for RD Sharma Maths Class 12 Chapter 8 Continuity Excercise 8.1 Question 10 Subquestion (viii)

Need solution for RD Sharma Maths Class 12 Chapter 8 Continuity Excercise 8.1 Question 10 Subquestion (viii)

Answers (1)

Answer:

                Continuous

Hint:

Continuous function must be defined at a point, limit must exist at the point, value of the function at that point must equal the value of the right and left  limit at that point.

Solution:

Given,

                f(x)=\left(\begin{array}{c} |x-a| \sin \left(\frac{1}{x-a}\right), \text { if } x \neq a \\\\ 0 \quad, \text { if } x=a \end{array}\right)

                f(x)=\left(\begin{array}{c} (x-a) \sin \left(\frac{1}{x-a}\right), \text { if } x>a \\\\ (-x+a) \sin \left(\frac{1}{-x+a}\right), \text { if } x<a \\\\ 0, \text { if } x=a \\ \end{array}\right)

We observe

[LHL at x=a]

                \lim _{x \rightarrow a^{-}} f(x)=(-a+a) \sin \left(\frac{1}{-a+a}\right)=0

[RHL at  x=a]

                \begin{aligned} &\lim _{x \rightarrow a^{+}} f(x)=(a-a) \sin \left(\frac{1}{a-a}\right)=0 \\\\ &\lim _{x \rightarrow a^{+}} f(x)=\lim _{x \rightarrow a^{-}} f(x)=f(a) \end{aligned}

Thus,  f\left ( x \right ) is continuous at  x=a.

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