document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=987818>Question 987818</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please explain this solution:\r<BR>\n" );
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document.write( "lim(x&#8594;0) sin(7x) cos(3x)/x <BR>\n" );
document.write( "= lim(x&#8594;0) cos(3x) * sin(7x)/x <BR>\n" );
document.write( "= lim(x&#8594;0) cos(3x) * 7sin(7x)/(7x) <BR>\n" );
document.write( "= cos 0 * (7 * 1), since lim(t&#8594;0) sin(t)/t = 1 with t = 7x <BR>\n" );
document.write( "= 7. \r<BR>\n" );
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document.write( "What causes cos(3x)*sin(7x)/x to turn into cos(3x)*7sin(7x)/(7x)? It looks like it is a common denominator of a fraction with sin but wouldn't that be invalid since it's sin(7x)/x isn't the sin(7x) internally connected and therefore cannot be broken into some like 7sin(7x)/7x. Or can it be thought of like sin/x*sin(7x)? Also, why doesn't cos(3x)/x become 3cos(3x)/3x or both of them together. Why is one chosen and the other gets left alone?\r<BR>\n" );
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document.write( "Thanks<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #608532 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=608532\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> What they did was multiply the fraction <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%287x%29%2Fx ALIGN=MIDDLE  ALT=\"sin%287x%29%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F7 ALIGN=MIDDLE  ALT=\"7%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is a fancy form of 1. Multiplying by this form of 1 does not change the expression\r<BR>\n" );
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document.write( "The reason why they did this was they wanted it in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28t%29%2Ft ALIGN=MIDDLE  ALT=\"sin%28t%29%2Ft\"   DISABLED_style= \"max-width:90%; height:auto;\" > where you replace the 't' with '7x'\r<BR>\n" );
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document.write( "All of this is done because normally the limit as x --> 0 has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%287x%29%2Fx ALIGN=MIDDLE  ALT=\"sin%287x%29%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > potentially undefined (because of that division by zero). There are no errors when you plug in x = 0 into the cos(3x) portion, so no algebraic tricks are needed there.\r<BR>\n" );
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document.write( "-------------------------------------------------------\r<BR>\n" );
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document.write( "If you need more help, or if you have any questions about the problem, feel free to email me at <font color=\"blue\"><code>jim_thompson5910@hotmail.com</code></font>  </SPAN>\n" );
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