Limit with Absolute

[nasi112]

I have a problem understanding limits with absolute.

[math]\lim_{x\to 0}\frac{2x - |x|}{|3x| - 2x}[/math]
If I compute the limit

[math]\lim_{x\to 0}\frac{2x - |x|}{|3x| - 2x} = \frac{0}{0}[/math]
This form is applicable for L'hopital rule. The problem I can't use the rule because there are absolute. What happens if I ignore the absolute and calculate the limit normally?

[math]\lim_{x\to 0}\frac{2x - x}{3x - 2x}[/math]

 

[blamocur]

The function [imath]f(x) = \frac{2x-|x|}{|3x|-2x}[/imath] is not continuous at 0. As is often the case, its limits are different for [imath]x<0[/imath] and [imath]x>0[/imath]. Have you learned about one-sided limits?

 

[Dr.Peterson]

What happens if I ignore the absolute and calculate the limit normally?

[math]\lim_{x\to 0}\frac{2x - x}{3x - 2x}[/math]

What you've done there is to assume x>0, so the limit you find is the limit from the right.

Now assume instead that x<0, rewrite the expression in that case (where |x| = -x), and find the limit from the left.

Then go to Desmos and graph the function, so see what is happening; and see how the graph relates to the work you did.