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Determine the Value of the Constant K So that the Function F ( X ) = { K X 2 , I F X ≤ 2 3 , I F X > 2 is Continuous at X = 2 . - Mathematics

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प्रश्न

Determine the value of the constant k so that the function

\[f\left( x \right) = \begin{cases}k x^2 , if & x \leq 2 \\ 3 , if & x > 2\end{cases}\text{is continuous at x} = 2 .\]

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उत्तर

\[f\left( x \right) = \binom{k x^2 , if x \leq 2}{3, if x > 2}\]

If

\[f\left( x \right)\] is continuous at x = 2, then 

\[\lim_{x \to 2^-} f\left( x \right) = \lim_{x \to 2^+} f\left( x \right) = f\left( 2 \right)\]

Now

\[\lim_{x \to 2^-} f\left( x \right) = \lim_{h \to 0} f\left( 2 - h \right) = \lim_{h \to 0} k \left( 2 - h \right)^2 = 4k\]

\[f\left( 2 \right) = 3\]

From (1), we have

\[4k = 3\]

\[ \Rightarrow k = \frac{3}{4}\]

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पाठ 9: Continuity - Exercise 9.1 [पृष्ठ १८]

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आरडी शर्मा Mathematics [English] Class 12
पाठ 9 Continuity
Exercise 9.1 | Q 21 | पृष्ठ १८

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