how do i compute the limit of |x|/x as x goes to 0?

Answer 1

For x >0, |x|/x = 1. So, lim (x --> 0 +) |x|/x = 1
For x <0, |x|/x = -1. So, lim (x --> 0 -) |x|/x = -1

So, this function has limits to the left and to the right of 0, but they are different. Therefore, |x|/x does not have a limit as x --> 0.

Answer 2

First look at the positive numbers (when x is really large and approaches 0).
|x| = x for x>0, so the function is x/x. Using l'Hopital's rule, you can see that this function approaches 1/1 = 1 from the positive side.

Now look at the negative numbers (x is a really large negative number and approaches 0).

|x| = -x in this case, so we have the function -x/x. Similarly, we can use l'Hopital's rule and see that this function approaches -1.

Since the function approaches -1 from x<0 and approaches 1 from x>0, there is no limit as x approaches 0.

Answer 3

divide by the higest power of x, in this case x. which leaves the ans as 1

Answer 4

sharky is right but you don't need l'hospital where he indicated