document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=219650>Question 219650</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I can't figure out how to simplify the expression (root x+2) - 3/(x-7). I'm actually in calculus I, and I have to take the limit of this expression as x goes to 7. I know I need to factor, but I don't know where to start. Any help would be very appreciated! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #165160 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=165160\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If we take the direct route, we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E7%2C%28sqrt%28x%2B2%29-3%29%2F%28x-7%29%29 ALIGN=MIDDLE  ALT=\"lim%28x-%3E7%2C%28sqrt%28x%2B2%29-3%29%2F%28x-7%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given limit.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%287%2B2%29-3%29%2F%287-7%29 ALIGN=MIDDLE  ALT=\"%28sqrt%287%2B2%29-3%29%2F%287-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=7 ALIGN=MIDDLE  ALT=\"x=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > (let's just pretend that we can evaluate this limit the intuitive way).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%289%29-3%29%2F%287-7%29 ALIGN=MIDDLE  ALT=\"%28sqrt%289%29-3%29%2F%287-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-3%29%2F%287-7%29 ALIGN=MIDDLE  ALT=\"%283-3%29%2F%287-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of 9 to get 3.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%29%2F%280%29 ALIGN=MIDDLE  ALT=\"%280%29%2F%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "Since the above is indeterminate, we have to take another approach...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E7%2C%28sqrt%28x%2B2%29-3%29%2F%28x-7%29%29 ALIGN=MIDDLE  ALT=\"lim%28x-%3E7%2C%28sqrt%28x%2B2%29-3%29%2F%28x-7%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given limit.\r<BR>\n" );
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document.write( "Because there's a radical in the numerator, we can try to rationalize the numerator in hopes that something will cancel.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E7%2C%28%28sqrt%28x%2B2%29-3%29%28sqrt%28x%2B2%29%2B3%29%29%2F%28%28x-7%29%28sqrt%28x%2B2%29%2B3%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both the numerator and denominator by the conjugate of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2B2%29-3 ALIGN=MIDDLE  ALT=\"sqrt%28x%2B2%29-3\"   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=lim%28x-%3E7%2C%28%28x%2B2%29-9%29%2F%28%28x-7%29%28sqrt%28x%2B2%29%2B3%29%29%29 ALIGN=MIDDLE  ALT=\"lim%28x-%3E7%2C%28%28x%2B2%29-9%29%2F%28%28x-7%29%28sqrt%28x%2B2%29%2B3%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL the numerator (use the difference of squares formula).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E7%2C%28x-7%29%2F%28%28x-7%29%28sqrt%28x%2B2%29%2B3%29%29%29 ALIGN=MIDDLE  ALT=\"lim%28x-%3E7%2C%28x-7%29%2F%28%28x-7%29%28sqrt%28x%2B2%29%2B3%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E7%2Ccross%28%28x-7%29%29%2F%28cross%28%28x-7%29%29%28sqrt%28x%2B2%29%2B3%29%29%29 ALIGN=MIDDLE  ALT=\"lim%28x-%3E7%2Ccross%28%28x-7%29%29%2F%28cross%28%28x-7%29%29%28sqrt%28x%2B2%29%2B3%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cancel out the common factors.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E7%2C1%2F%28sqrt%28x%2B2%29%2B3%29%29 ALIGN=MIDDLE  ALT=\"lim%28x-%3E7%2C1%2F%28sqrt%28x%2B2%29%2B3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. Now let's see if we can evaluate this limit using the naive approach (ie use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3Ea%2Cf%28x%29%29=f%28a%29 ALIGN=MIDDLE  ALT=\"lim%28x-%3Ea%2Cf%28x%29%29=f%28a%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=1%2F%28sqrt%287%2B2%29%2B3%29 ALIGN=MIDDLE  ALT=\"1%2F%28sqrt%287%2B2%29%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=7 ALIGN=MIDDLE  ALT=\"x=7\"   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=1%2F%28sqrt%289%29%2B3%29 ALIGN=MIDDLE  ALT=\"1%2F%28sqrt%289%29%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%283%2B3%29 ALIGN=MIDDLE  ALT=\"1%2F%283%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of 9 to get 3.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lim%28x-%3E7%2C%28sqrt%28x%2B2%29-3%29%2F%28x-7%29%29=1%2F6 ALIGN=MIDDLE  ALT=\"lim%28x-%3E7%2C%28sqrt%28x%2B2%29-3%29%2F%28x-7%29%29=1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "An alternative (and better way in my opinion) is to use L' Hospital's Rule. If you haven't learned that yet, then ignore that last suggestion (note: the rule only works if you're dealing with interderminate forms). </SPAN>\n" );
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