Question

If $f\left(z\right)=\{\begin{array}{cc}6z,& z\le -4\\ -12-9z,& z>-4\end{array}$, then the value of the $\underset{z\to -4}{lim}f\left(z\right)$ is

• A. $-24$
• B. $24$
• C. Does not exist.
• D. $-4$

Answer 1

The correct option is C Does not exist.

$L.H.L.=\underset{z\to -4^{-}}{lim}f\left(z\right)=\underset{z\to -4^{-}}{lim}6z$
$=\underset{h\to 0}{lim}6(-4-h)=6\times (-4)$
$=-24$
$(\because z\to -4^{-}$ implies that $z<-4)$

$R.H.L.=\underset{z\to -4^{+}}{lim}f\left(z\right)=\underset{z\to -4^{+}}{lim}(-12-9z)$
$=\underset{h\to 0}{lim}(-12-9(-4+h\left)\right)=-12-9\times (-4)$
$=24$
$(\because z\to -4^{+}$ implies that $z>-4)$

Clearly,
$L.H.L.\ne R.H.L.$
So, the limit does not exist.