Limits at a point

Quote:

Originally Posted by Glitch

Hey guys,

I'm trying to understand how to determine the left and right side of this limit:

limit as x -> 2

|x-2|/(x-2)

Do I have to graph it, or is there an algebraic way to do this? Cheers.

Recall that

$|x - 2| = \begin{cases}\phantom{-(}x-2\phantom{)}\textrm{ if }x-2\geq 0\\-(x - 2)\textrm{ if }x-2<0\end{cases}$

$|x - 2| = \begin{cases}x-2\textrm{ if }x\geq 2\\2-x\textrm{ if }x<2\end{cases}$

When you are are approaching $x = 2$ from the left, $x < 2$ .

So $\lim_{x \to 2^{-}}\frac{|x-2|}{x-2} = \lim_{x\to 2^{-}}\frac{2-x}{x-2}$

$= \lim_{x\to 2^{-}}\frac{-(x-2)}{x-2}$

$= \lim_{x\to 2^{-}}(-1)$

$= -1$ .

When you are approaching $x=2$ from the right, $x>2$ .

So $\lim_{x \to 2^{+}}\frac{|x-2|}{x-2} = \lim_{x\to 2^{+}}\frac{x-2}{x-2}$

$=\lim_{x \to 2^{+}}(1)$

$= 1$ .

Clearly, since the left hand limit is not the same as the right hand limit, the limit does not exist.