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  • Need solution for RD Sharma Maths Class 12 Chapter 8 Continuity Excercise 8.1 Question 23

Need solution for RD Sharma Maths Class 12 Chapter 8 Continuity Excercise 8.1 Question 23

Answers (1)

Answer:

a=-2

Hint:

f\left ( x \right )  must be defined. The limit of the f\left ( x \right ) approaches the value x must exist.

Given:

                f(x)=\left(\begin{array}{r} a x+5, \text { if } x \leq 2 \\\\ x-1, \text { if } x>2 \end{array}\right) 

Solution:

                f(x)=\left(\begin{array}{r} a x+5, \text { if } x \leq 2 \\\\ x-1, \text { if } x>2 \end{array}\right)

We observe

[LHL at x=2 ]

                \lim _{x \rightarrow 2^{-}} f(x)=\lim _{h \rightarrow 0} a(2-h)+5=2 a+5

[RHL at x=2 ]

                \lim _{x \rightarrow 2^{+}} f(x)=\lim _{h \rightarrow 0} f(2+h)=\lim _{h \rightarrow 0}(2+h-1)=1

f(2)=a(2)+5=2 a+5

f\left ( x \right )   is continuous at x=2, we have

                \begin{aligned} &\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)=f(2) \\\\ &2 a+5=1 \\\\ &2 a=-4 \\\\ &a=-2 \end{aligned}

 

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