Proving y=ln|x| and its Limits: A Comprehensive Guide

[Ayinajin]

I’ll appreciated very much if anyone can help me with any of these following proves by any means. Thank you!

prove:
1. y=ln|x| = y'=1/x

2. lim ln|x|=+ infinity as x approaches toward positive infinity and-
lim ln|x| = - infinity as x approaches 0 from positive side

 

[Muzza]

Maybe something like this... Let f(x) = exp(ln(x)), where exp(x) = e^x. Using the chain rule, we have that f'(x) = exp(ln(x)) * d/dx(ln(x)) = x * d/dx(ln(x)). But since f(x) = x, we must also have f'(x) = 1. Thus x * d/dx(ln(x)) = 1, or d/dx(ln(x)) = 1/x.

 

[Ayinajin]

Maybe something like this... Let f(x) = exp(ln(x)), where exp(x) = e^x. Using the chain rule, we have that f'(x) = exp(ln(x)) * d/dx(ln(x)) = x * d/dx(ln(x)). But since f(x) = x, we must also have f'(x) = 1. Thus x * d/dx(ln(x)) = 1, or d/dx(ln(x)) = 1/x.

Thank you Muzza! That was beautiful! How about my second problem?