Sketching Curves

Date: 6/24/96 at 15:31:5
From: Asim Razzaq
Subject: Sketching Curves

Question: Sketch the curve, y = log natural of x / (x-1)
                             
                            i.e log natural of x divided by x-1

Date: 6/24/96 at 19:38:48
From: Doctor Anthony
Subject: Re: Sketching Curves 

The curve is not defined for x < 0.

When x -> 0 ln(x) -> -infinity, and dividing by -1, we see that the 
curve approaches +infinity as x ->0.  When x=1, y=0/0, so use 
l'Hopital's rule, i.e. expression is equal to the differential of the 
numerator divided by the differential of the denominator.  

At x=1 the expression becomes 

(1/x)/1 = 1/x = 1/1 = 1

So the curve passes through (1,1). As x increases to infinity the 
expression tends to 1/x (again by l'Hopital's rule), and so 
y->0.  The general shape is therefore a curve starting at + infinity 
when x=0, descending through the point (1,1) and becoming asymptotic 
to the x axis as x tends to infinity.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/