Question

If $h\left(z\right)=\{\begin{array}{cc}6z,& z\le -4\\ 1-9z,& z>-4\end{array}$, then value of the $\underset{z\to -4}{lim}h\left(z\right)=$

• A. $-24$
• B. $37$
• C. It does not exist
• D. $-4$

Answer 1

The correct option is C It does not exist

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

$R.H.L.=\underset{z\to -4^{+}}{lim}h\left(z\right)=\underset{z\to -4^{+}}{lim}(1-9z)=1-9\times (-4)=37$
$(\because z\to -4^{+}$ implies that $z>-4)$

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