Question

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

Answer 1

use one-sided limits

Answer 2

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

Answer 3

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

Answer 4

Exactly

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

you could change the abs value to
x-2 for x>2

2-x for x<2

take limit of each case

Answer 9

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

Answer 10

no it does not

Answer 11

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