First slide

Introduction to limits

Question

$lim_{x \to 2} \frac{x - 2}{| x - 2 |}$ is equal to

Easy

A

1
B

-1
C

0
D

Dose not exist

Solution

$∵ | x - 2 | = \{\begin{cases} x - 2, if x \geq 2 \\ - (x - 2), if x < 2 \end{cases}$
Now, RHL $= lim_{x \to 2^{+}} (\frac{x - 2}{x - 2}) = lim_{x \to 2^{+}} 1 = 1$
and LHL $= lim_{x \to 2^{-}} (\frac{x - 2}{- (x - 2)})$
$= - lim_{x \to 2^{-}} (\frac{x - 2}{x - 2}) = - 1$
since $lim_{x \to 2^{+}} f (x) \neq lim_{x \to 2^{-}} f (x)$
Therefore, the limit does not exist.

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