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

$a=-2$

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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