Sketching CurvesDate: 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/ |
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