# Let F(X) Be a Function Defined by F ( X ) = { 3 X | X | + 2 X , X ≠ 0 0 , X = 0 . Show that Lim X → 0 F ( X ) Does Not Exist. - Mathematics

Let f(x) be a function defined by $f\left( x \right) = \begin{cases}\frac{3x}{\left| x \right| + 2x}, & x \neq 0 \\ 0, & x = 0\end{cases} .$ Show that $\lim_{x \to 0} f\left( x \right)$ does not exist.

#### Solution

$f\left( x \right) = \begin{cases}\frac{3x}{\left| x \right| + 2x}, & x \neq 0 \\ 0, & x = 0\end{cases}$
$\text{ Left hand limit }:$
$\lim_{x \to 0^-} \left[ \frac{3x}{\left| x \right| + 2x} \right]$
$\text{ Let } x = 0 - h, \text{ where } h \to 0 .$
$\Rightarrow \lim_{h \to 0} \left[ \frac{3\left( - h \right)}{\left| - h \right| + 2\left( - h \right)} \right]$
$= \lim_{h \to 0} \left[ \frac{- 3h}{h - 2h} \right]$
$= \lim_{h \to 0} \left[ \frac{- 3h}{- h} \right]$
$= 3$
$\text{ Right hand limit }:$
$\lim_{x \to 0^+} \left( \frac{3x}{\left| x \right| + 2x} \right)$
$\text{ Let } x = 0 + h, \text{ where } h \to 0 .$
$\Rightarrow^{} \lim_{h \to 0} \left( \frac{3h}{\left| h \right| + 2h} \right)$
$= \lim_{h \to 0} \left( \frac{3h}{h + 2h} \right)$
$= 1$
$\lim_{x \to 0^-} \left( \frac{3x}{\left| x \right| + 2x} \right) \neq \lim_{x \to 0^+} \left( \frac{3x}{\left| x \right| + 2x} \right)$
$\text{ Thus }, \lim_{x \to 0} f\left( x \right) \text{ does not exist } .$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.1 | Q 4 | Page 11