Verified answer

start with what's inside the parentheses: what does the function x/(1 + x^2) look like? That is, what is its range? You find that the answer to that question is that it is constantly between -1 and 1, spanning every number in between. An algebraic "reason" would be that |x|<|1 + x^2| for all real numbers, but short of that I don't have an intuitive explanation.

Well, once you've figured that out, you'll know that |x/(1 + x^2)| is strictly between 0 and 1 (again, spanning everything in between) simply because anything negative becomes positive.

The last part would be to figure out what happens to the range when you take the log. The best way to figure this out is by taking the ln of the "edges" of the range of whatever is inside of the log. The highest |x/(1+ x²)| goes to is 1, so ln(1) = 0 would be the upper limit for the range of the total function.

On the other side, the lowest it goes to is 0, so ln(0) =(that is, it approaches) -∞ would be the lower limit on the range of the total function.

hope that helps

http://www.wolframalpha.com/input/?i=limit+y%3Dln+... will compute and display the limits for you. But it does not show the steps.