Question

Calculate the limit of the following function if it exists

\lim\limits_{x\to 0} \dfrac{|x|}{x}

Collected in the board: Limit

Steven Zheng posted 1 month ago

Answer


Remove the absolute value sign of the numerator, we get a piecewise function.


\begin{cases} -x &\text{if } x< 0 \\ x &\text{if } x> 0 (x\ne 0) \end{cases}

So let's calculate the left and right limits separately.


Left limit,

\lim\limits_{x\to 0^-} \dfrac{-x}{x} = -1


Right limit,

\lim\limits_{x\to 0^+} \dfrac{x}{x} = 1


We get different values. So the limit of the function does not exist since the limit exists if and only if left and right limits exist and are equal.

Steven Zheng posted 1 month ago

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