Determine the value of the constant ' k ' so that the function f(… - Vidyadip

Question

Determine the value of the constant ' $k$ ' so that the function $f(x)=\left\{\begin{array}{cll}\frac{k}{|x|}, & \text { if } & x<0 \\ 3, & \text { if } & x \geq 0\end{array}\right.$ is continuous at $x=0$.

✓

Answer

Since $f$ is continuous at $x=0$,$
\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=f(0)
$
Here
$
\begin{aligned}
f(0) & =3, \\
LHL & =\lim _{x \rightarrow 0^{-}} f(x)
\end{aligned}
$
$=\lim _{x \rightarrow 0^{-}} \frac{k x}{|x|}=\lim _{x \rightarrow 0^{-}} \frac{k x}{-x}=-k$
$\therefore \quad-k=3$ or $k=-3$.

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