The limit of absolute value function?

Calculus Level 1

$\Large \lim_{x \to 0} \frac{x}{|x|} = \ ?$

1

Undefined (No limit exists)

\pm 1

-1

1 solution

Michael Huang
Dec 2, 2016

Left-Hand Side Limit

At $x < 0$ , $|x| = -x$ . Then, $\lim_{x\rightarrow 0^{-}} \dfrac{x}{|x|} = \lim_{x\rightarrow 0} \dfrac{x}{-(x)} = -1$

Right Hand Side Limit

At $x > 0$ , $|x| = x$ . Then, $\lim_{x\rightarrow 0^{+}} \dfrac{x}{|x|} = \lim_{x\rightarrow 0} \dfrac{x}{x} = 1$

Answer

Since $\lim_{x\rightarrow 0^{-}} \dfrac{x}{|x|} \neq \lim_{x\rightarrow 0^{+}} \dfrac{x}{|x|}$ this concludes that $\boxed{\text{limit does not exist}}$ .

The limit of absolute value function?

1 solution

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