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Sketching Curves

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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
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```
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/
```
Associated Topics:
High School Equations, Graphs, Translations

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