Question

For $\underset{x\to 0}{lim}\frac{|2x|}{x}$, which of the following is/are correct?

• A. The right hand limit is $2.$
• B. The left hand limit is $-2.$
• C. The limit doesn't exist.
• D. The left hand limit is $2.$

Answer 1

Answer: (A), (B), (C)

The limit doesn't exist.

Given : $\underset{x\to 0}{lim}\frac{|2x|}{x}=2\underset{x\to 0}{lim}\frac{|x|}{x}$
$L.H.L.=2\underset{x\to 0^{-}}{lim}\frac{|x|}{x}$
$=2\underset{h\to 0}{lim}\frac{|0-h|}{0-h}$
$=2\underset{h\to 0}{lim}\frac{h}{-h}=-2$

$R.H.L.=2\underset{x\to 0^{+}}{lim}\frac{|x|}{x}$
$=2\underset{h\to 0}{lim}\frac{|0+h|}{0+h}$
$=2\underset{h\to 0}{lim}\frac{h}{h}=2$
Limit doesn't exist.

Alternate Solution:
Drawing the graph of $f\left(x\right)=\frac{|2x|}{x}$


From the graph of $f\left(x\right)$, we can say that Limit at $x=0$ doesn't exist.