What is the limit of abs(x-2)/(x-2) as x approaches 2?

Question

helder_edwin · Answer

use one-sided limits

anonymous · Answer

abs(x-2)/(x-2) means just the absolute value of (x-2) divided by (x-2).

helder_edwin · Answer

$$ \large \lim_{x\to2}\frac{|x-2|}{x-2} $$
right?

anonymous · Answer

Exactly

helder_edwin · Answer

compute separatedly
$$ \Large \lim_{x\to2+}\frac{|x-2|}{x-2} $$
and
$$ \Large \lim_{x\to2-}\frac{|x-2|}{x-2} $$
(one-sided limits)

anonymous · Answer

Alright, I understand the graphical process of one-sided limits, but is there an analytical (algebraic) approach?

helder_edwin · Answer

just follow this:
$$ \Large x\to2+\Rightarrow x>2\Rightarrow x-2>0\Rightarrow
    |x-2|=x-2 $$

dumbcow · Answer

you could change the abs value to
x-2  for x>2

2-x  for x<2

take limit of each case

anonymous · Answer

Does the limit here actually exist? Since the limit from the left isn't equal to the limit from the right.

helder_edwin · Answer

no it does not

anonymous · Answer

So would the answer be "does not exist b/c..." or the two different values it approaches from either side?