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Find the Values of a So that the Function F ( X ) = { a X + 5 , I F X ≤ 2 X − 1 , I F X > 2 is Continuous at X = 2 . - Mathematics

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प्रश्न

Find the values of a so that the function 

\[f\left( x \right) = \begin{cases}ax + 5, if & x \leq 2 \\ x - 1 , if & x > 2\end{cases}\text{is continuous at x} = 2 .\]
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उत्तर

Given:

\[f\left( x \right) = \binom{ax + 5, if x \leq 2}{x - 1, if x > 2}\]
 We observe
(LHL at x = 2) = 
\[\lim_{h \to 0} a\left( 2 - h \right) + 5 = 2a + 5\]
(RHL at x = 2) = \[\lim_{x \to 2^+} f\left( x \right) = \lim_{h \to 0} f\left( 2 + h \right)\]
\[\lim_{h \to 0} \left( 2 + h - 1 \right)\]
\[1\]
And,
\[f\left( 2 \right) = a\left( 2 \right) + 5 = 2a + 5\]
Since
\[f\left( x \right)\]  is continuous at x = 2, we have
\[\lim_{x \to 2^-} f\left( x \right) = \lim_{x \to 2^+} f\left( x \right) = f\left( 2 \right)\]
\[2a + 5 = 1\]
\[2a = - 4\]
\[a = - 2\]
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पाठ 9: Continuity - Exercise 9.1 [पृष्ठ १९]

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आरडी शर्मा Mathematics [English] Class 12
पाठ 9 Continuity
Exercise 9.1 | Q 23 | पृष्ठ १९

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