# calculus

how do I find the range of a natural log like
ln | x/(1+x^2) |

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1. Consider the function
u = │x/(1+x^2)│

it looks like birdwings, with (0,0) a cusp, the curve rising to a maximum of (1,1/2) and then dropping again to approach the x-axis with the x-axis as an asymptote. It is symmetric about the y-axis.

so then you are looking at ln u

u lies between 0 and 1/2 , so ln u will always be negative.

ln 1/2 = -.693 and for all other of u values which will be less than 1/2 ln u will be less than -.693

as u --> 0 ln u approaches negative infinity

so the range is ≤ -.693

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