# Show that Lim X → 0 X | X | Does Not Exist. - Mathematics

Show that $\lim_{x \to 0} \frac{x}{\left| x \right|}$ does not exist.

#### Solution

$\lim_{x \to 0} \left( \frac{x}{\left| x \right|} \right)$Left hand limit:

$\lim_{x \to 0^-} \left( \frac{x}{\left| x \right|} \right)$
$\text{ Let } x = 0 - h, \text{ where } h \to 0 .$
$\Rightarrow \lim_{h \to 0} \left( \frac{0 - h}{\left| 0 - h \right|} \right)$
$= \lim_{h \to 0} \left( \frac{- h}{h} \right)$
$= - 1$

Right hand limit:

$\lim_{x \to 0^+} \frac{\left( x \right)}{\left| x \right|}$
$\text{Let } x = 0 + h, \text{ where } h \to 0 .$
$\lim_{h \to 0} \left( \frac{0 + h}{\left| 0 + h \right|} \right)$
$= \lim_{h \to 0} \left( \frac{h}{h} \right)$
$= 1$

Left hand limit ≠ Right hand limit $Thus, \lim_{x \to 0} \left( \frac{x}{\left| x \right|} \right) \text{ does not exist } .$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.1 | Q 1 | Page 11