Question

Which of the following option(s) is (are) true for $\underset{x\to 0}{lim}\frac{1}{\ln |x|}$

• A. limit exists
• B. $L.H.L.=1$
• C. $R.H.L.=0$
• D. limit does not exist

Answer 1

Answer: (A), (C)

$R.H.L.=0$

Given: $\underset{x\to 0}{lim}\frac{1}{\ln |x|}$
$L.H.L.=\underset{x\to 0^{-}}{lim}\frac{1}{\ln |x|}=\underset{h\to 0}{lim}\frac{1}{\ln |0-h|}$
$=\underset{h\to 0}{lim}\frac{1}{\ln \left(h\right)}=0$
$R.H.L.=\underset{x\to 0^{+}}{lim}\frac{1}{\ln |x|}=\underset{h\to 0}{lim}\frac{1}{\ln |0+h|}$
$=\underset{h\to 0}{lim}\frac{1}{\ln \left(h\right)}=0$
$\Rightarrow L.H.L=R.H.L.$
$\therefore$ Limit exists.