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Question

Evaluate $$\displaystyle \lim_{x\rightarrow 0}f\left ( x \right ),$$ where $$f(x) = \displaystyle \begin{cases}\cfrac{\left | x \right |}{x} ,& \text{ } x\neq  0 \\0, & \text{ } x= 0\ \, \end{cases}$$


Solution

$$f(x) = \displaystyle \begin{cases}\cfrac{\left
| x \right |}{x} ,& \text{ } x\neq  0 \\0, & \text{ } x= 0\ \,
\end{cases}$$
$$\displaystyle \lim_{x\rightarrow 0^{-}}f\left ( x
\right )= \lim_{x\rightarrow 0}\left (
\frac{-x}{x} \right )=-1$$
$$\displaystyle
\lim_{x\rightarrow 0^{+}}f\left ( x \right )=\lim_{x\rightarrow 0}\left ( \frac{x}{x} \right )=1$$
Clearly  $$\displaystyle \lim_{x\rightarrow 0^{-}}f\left ( x
\right )\neq \lim_{x\rightarrow 0^{+}}f\left ( x \right )\cdot $$
Hence $$\displaystyle \lim_{x\rightarrow 0}f\left ( x \right )$$ does not exist.

Mathematics
NCERT
Standard XI

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