Solve this


If $f(x)=\left\{\begin{array}{ll}2 x^{2}+k, & \text { if } x \geq 0 \\ -2 x^{2}+k, & \text { if } x<0\end{array}\right.$, then what should be the value of

$k$ so that $f(x)$ is continuous at $x=0$.


The given function can be rewritten as

$(x)=\left\{\begin{array}{l}2 x^{2}+k, \text { if } x \geq 0 \\ -2 x^{2}+k, \text { if } x<0\end{array}\right.$

We have

$(\mathrm{LHL}$ at $x=0)=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(-h)=\lim _{h \rightarrow 0}-2(-h)^{2}+k=k$

$(\mathrm{RHL}$ at $x=0)=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} f(0+h)=\lim _{h \rightarrow 0} f(h)=\lim _{h \rightarrow 0}\left(2 h^{2}+k\right)=k$

If $f(x)$ is continuous at $x=0$, then

$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)$

$\Rightarrow \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=k$

∴ can be any real number.

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