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← Revision 18 as of 2008-11-14 13:42:04 ⇥
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http://moinmoin.wikiwikiweb.de/ [http://www.python.org/ Python] | http://moinmoin.wikiwikiweb.de/ [[http://www.python.org/|Python]] |
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== A graphical illustration == x=var('x') @interact def _(x = slider(-7/10,7/10,1/20,1/2)): html('<h3>A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$</h3>') html('Below is a unit circle, so the length of the <font color=red>red line</font> is |sin(x)|') html('and the length of the <font color=blue>blue line</font> is |tan(x)| where x is the length of the arc.') html('is |sin(x)| and |tan(x)|, respectively where x is length of the arc.') html('From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|') html('and it follows easily from this that') html('cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.') html('As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.') if not (x == 0): pretty_print("sin(x)/x = "+str(sin(float(x))/float(x))) elif x == 0: pretty_print("The limit of sin(x)/x as x tends to 0 is 1") C=circle((0,0),1, rgbcolor='black') mvp = (cos(x),sin(x));tpt = (1, tan(x)) p1 = point(mvp, pointsize=30, rgbcolor='red'); p2 = point((1,0), pointsize=30, rgbcolor='red') line1 = line([(0,0),tpt], rgbcolor='black'); line2 = line([(cos(x),0),mvp], rgbcolor='red') line3 = line([(0,0),(1,0)], rgbcolor='black'); line4 = line([(1,0),tpt], rgbcolor='blue') result = C+p1+p2+line1+line2+line3+line4 result.show(aspect_ratio=1, figsize=[3,3], axes=False) |
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