document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1131261>Question 1131261</A>: <I><SPAN STYLE=\"word-break: break-all;\"> PROVING<BR>\n" );
document.write( "A. limit as delta x approaches to zero   cos delta x-1/delta x<BR>\n" );
document.write( "B. limit as delta x approaches to zero   sin delta x/delta x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #747891 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=747891\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Segment SC = <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sin(x)\"> by definition (note unit circle).  Altitude of triangle OSU is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sin(x)\"> and the base of triangle OSU is 1, so the area of triangle OSU is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{\sin(x)}{2}\">.\r<BR>\n" );
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document.write( "The area of a circle sector is given by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{\theta{r^2}}{2}\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \theta\"> is the subtended angle.  Hence, the area of sector OSU is simply <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{x}{2}\">\r<BR>\n" );
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document.write( "Segment TU = <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \tan(x)\"> by definition of the tangent function. (TU divided by OU, but OU measures 1).  Then the area of triangle OTU is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{\tan(x)}{2}\">.\r<BR>\n" );
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document.write( "From the diagram, the area of triangle OSU is less than or equal to the area of sector OSU which in turn is less than or equal to triangle OTU.  Hence:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\sin(x)}{2}\ \leq\ \frac{x}{2}\ \leq\ \frac{\tan(x)}{2}\">\r<BR>\n" );
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document.write( "Multiply by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{2}{\sin(x)}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 1\ \leq\ \frac{x}{\sin(x)}\ \leq\ \frac{1}{\cos(x)}\">\r<BR>\n" );
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document.write( "Take the reciprocal and reverse the sense of the inequalities:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \cos(x)\ \leq\ \frac{\sin(x)}{x}\ \leq\ 1\">\r<BR>\n" );
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document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \lim_{x\right{0}}\,\cos(x)\ =\ 1\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \lim_{x\right{0}}\,1\ =\ 1\">\r<BR>\n" );
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document.write( "Therefore, by the Squeeze Theorem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \lim_{x\right{0}}\,\frac{\sin(x)}{x}\ =\ 1\"><BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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