Question about absolute value limits?

[emlekarc]

When do you check the limit from the right and left of a limit with an absolute value in the numerator or denominator?

For example, why do you check the limit from both sides of:

Lim x -> 3/2 (2x^2-3x)/absolute value(2x-3)

But only the left side of:

limit as x approaches -2

(2-absolute value x)/(2+x)

 

[Office_Shredder]

If the thing inside the absolute value is equal to zero when you plug in the limiting value of x, then depending on whether x is a little bigger or a little smaller it will change how the absolute value acts on the expression inside of it. For example

$$\lim_{x\to 3/2} \frac{...}{|2x-3|}$$
if x = 3/2, 2x-3 = 0. So if x > 3/2, |2x-3| = 2x-3, but if x < 3/2, |2x-3| = 3-2x and you need to be careful about this distinction.

On the other hand, if x = -2, 2-x = 4. So if x > -2 by a little bit, |2-x| = 2-x, but if x < -2 by a little bit, |2-x| = 2-x still. In both cases you get the same expression when you drop the absolute value sign.