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Question
If \[f\left( x \right) = \begin{cases}2 x^2 + k, &\text{ if } x \geq 0 \\ - 2 x^2 + k, & \text{ if } x < 0\end{cases}\] then what should be the value of k so that f(x) is continuous at x = 0.
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Solution
The given function can be rewritten as
We have
(LHL at x = 0) =
(RHL at x = 0) =\[\lim_{x \to 0^+} f\left( x \right) = \lim_{h \to 0} f\left( 0 + h \right) = \lim_{h \to 0} f\left( h \right) = \lim_{h \to 0} \left( 2 h^2 + k \right) = k\]
If
\[ \Rightarrow \lim_{x \to 0^-} f\left( x \right) = \lim_{x \to 0^+} f\left( x \right) = k\]
∴ k can be any real number.
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