document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174500>Question 1174500</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following limit:<BR>\n" );
document.write( " lim x→4     <BR>\n" );
document.write( "(x^2 − 4x/ x^ 2 − 16) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #799921 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=799921\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "For a moment, let's completely ignore the limit portion or limit notation.\r<BR>\n" );
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document.write( "Let's focus on the expression only. So we'll focus on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-4x%29%2F%28x%5E2-16%29 ALIGN=MIDDLE  ALT=\"%28x%5E2-4x%29%2F%28x%5E2-16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Simplifying the expression leads to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%28x-4%29%29%2F%28x%5E2-16%29 ALIGN=MIDDLE  ALT=\"%28x%28x-4%29%29%2F%28x%5E2-16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%28x-4%29%29%2F%28%28x-4%29%28x%2B4%29%29 ALIGN=MIDDLE  ALT=\"%28x%28x-4%29%29%2F%28%28x-4%29%28x%2B4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Ahighlight%28%28x-4%29%29%29%2F%28highlight%28%28x-4%29%29%28x%2B4%29%29 ALIGN=MIDDLE  ALT=\"%28x%2Ahighlight%28%28x-4%29%29%29%2F%28highlight%28%28x-4%29%29%28x%2B4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Across%28%28x-4%29%29%29%2F%28cross%28%28x-4%29%29%28x%2B4%29%29 ALIGN=MIDDLE  ALT=\"%28x%2Across%28%28x-4%29%29%29%2F%28cross%28%28x-4%29%29%28x%2B4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F%28x%2B4%29 ALIGN=MIDDLE  ALT=\"x%2F%28x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Such a simplification is necessary because plugging x = 4 into the original expression leads to a division by zero error<BR>\n" );
document.write( "Note how x^2-16 = 4^2 - 16 = 16 - 16 = 0 would be in the denominator if x = 4.\r<BR>\n" );
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document.write( "But with the simplified expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F%28x%2B4%29 ALIGN=MIDDLE  ALT=\"x%2F%28x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can plug in x = 4 just fine<BR>\n" );
document.write( "Doing so gets us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F%28x%2B4%29+=+4%2F%284%2B4%29+=+4%2F8+=+1%2F2 ALIGN=MIDDLE  ALT=\"x%2F%28x%2B4%29+=+4%2F%284%2B4%29+=+4%2F8+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "As x gets closer and closer to x = 4, the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x%2F%28x%2B4%29 ALIGN=MIDDLE  ALT=\"y+=+x%2F%28x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gets closer and closer to y = 1/2 as we apply the limit. As with all limits, we don't actually arrive at the exact value. We only get closer. \r<BR>\n" );
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document.write( "Answer: 1/2\r<BR>\n" );
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