1. ## limits with radicals and with absolute values

I have two limit questions:

a) lim x goes to 1 of ((square root (6-x) -2)/ (square root (10-x) -3))
b) lim x goes to 0 of (2x / (abs(3x-2) - abs(3x+2))

The first one I tried to multiply by both conjugates but it doesn't result in things canceling like it should. The second one I can't work out the intervals. I graphed it and see the limit is a negative third but I can't see that algebraically.

2. ## Re: limits with radicals and with absolute values

"Things don't cancel like they should" because the numerator is NOT 0 when x= 1. What does that tell you about the limit?

As for the second, |x| "changes formula" at x= 0. |3x- 2| changes formula when 3x- 2= 0 and |3x+2| changes formula when 3x+ 2= 0. What are the solutions to that? Those values of x divide the real number line into three intervals.

3. ## Re: limits with radicals and with absolute values

But the denominator is zero and 1/0 is undefined. So is do I just say the limit is undefined? That doesn't seem like much of a question then.

I tried setting it up with intervals of less than -2/3, -2/3 < x < 0, 0<x<2/3 and greater than 2/3 but then for the limit to exist all of the intervals should give me the same value. But when I worked it out for x < -2/3 both abs give negative results and the denominator become 4 rather than the -6x I want to get a value of -1/3.