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In the following, determine the value of constant involved in the definition so that the given function is continuou: \[f\left( x \right) = \begin{cases}k( x^2 + 3x), & \text{ if }  x < 0 \\ \cos 2x , - Mathematics

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Question

In the following, determine the value of constant involved in the definition so that the given function is continuou:  \[f\left( x \right) = \begin{cases}k( x^2 + 3x), & \text{ if }  x < 0 \\ \cos 2x , & \text{ if }  x \geq 0\end{cases}\]

Sum
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Solution

Given:

 \[f\left( x \right) = \begin{cases}k( x^2 + 3x), & \text{ if }  x < 0 \\ \cos 2x , & \text{ if }  x \geq 0\end{cases}\]
If  \[f\left( x \right)\]  is continuous at x = 0, then 
\[\lim_{x \to 0^-} f\left( x \right) = \lim_{x \to 0^+} f\left( x \right)\]
\[\Rightarrow \lim_{h \to 0} f\left( - h \right) = \lim_{h \to 0} f\left( h \right)\]
\[ \Rightarrow \lim_{h \to 0} \left( k\left( \left( - h \right)^2 - 3h \right) \right) = \lim_{h \to 0} \left( \cos 2h \right)\]
\[ \Rightarrow 0 = 1 \left[ \text{It is not possible} \right]\]
Hence, there does not exist any value of k, which can make the given function continuous.
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Chapter 9: Continuity - Exercise 9.2 [Page 35]

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RD Sharma Mathematics [English] Class 12
Chapter 9 Continuity
Exercise 9.2 | Q 4.3 | Page 35

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