document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=969699>Question 969699</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Lim           Sqrt{4x+1}-3 / x-2<BR>\n" );
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document.write( "How do i do this. I cant figure out how to get rid of the sqrt. I looked online and everyone says to rationalize the numerator but they skip steps so idk what I'm supposed to do. Can someone explain without skipping a step even if it seems easy to you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #592503 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=592503\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> find the limit of sqrt(4x+1)-3 / (x-2) as x approaches 2.\r<BR>\n" );
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document.write( "i believe that what they told you is a correct way to look at it.\r<BR>\n" );
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document.write( "if you were able to graph the function, you would have found that the answer should be 2/3 or thereabouts.\r<BR>\n" );
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document.write( "that's just by eyeballing.\r<BR>\n" );
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document.write( "algebraically, you would do the following:\r<BR>\n" );
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document.write( "start with:\r<BR>\n" );
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document.write( "(sqrt(4x+1)-3) / (x-2)\r<BR>\n" );
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document.write( "multiply numerator and denominator by (sqrt(4x+1)+3) to get:\r<BR>\n" );
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document.write( "((sqrt(4x+1)-3)*(sqrt(4x+1)+3)) / ((x-2)*(sqrt(4x+1)+3))\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "((4x+1)-9) / ((x-2)*(sqrt(4x+1)+3))\r<BR>\n" );
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document.write( "simplify further to get:\r<BR>\n" );
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document.write( "(4x-8) / ((x-2)*(sqrt(4x+1)+3))\r<BR>\n" );
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document.write( "factor out a 4 in the numerator to get:\r<BR>\n" );
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document.write( "4*(x-2) / ((x-2)*(sqrt(4x+1)+3))\r<BR>\n" );
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document.write( "the (x-2) in the numerator and denominator cancel out and you are left wtih:\r<BR>\n" );
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document.write( "4 / (sqrt(4x+1)+3))\r<BR>\n" );
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document.write( "when x = 2, this becomes 4 / (sqrt(9)+3)) which becomes 4 / (3+3) which becomes 4/6 which becomes 2/3.\r<BR>\n" );
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document.write( "that's your solution.\r<BR>\n" );
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document.write( "the limit of (sqrt(x+1)-3) / (x-2) as x approaches 2 is equal to 2/3.\r<BR>\n" );
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document.write( "in the graph, you can see that there is a hole at x = 2, and if you gave values of x as 1.999999999 and 2.000000001, you would see that they hovered about 2/3.\r<BR>\n" );
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document.write( "the graph looks like this:\r<BR>\n" );
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document.write( "the horizontal line is at y = 2/3.\r<BR>\n" );
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document.write( "you can see that the graph of y = 2/3 intersects with the graph of y = (sqrt(4x+1)-3)/(x-2) at x = 2.\r<BR>\n" );
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document.write( "you will not, however, be able to find that value since the function is undefined at x = 2.  \r<BR>\n" );
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document.write( "there is a hole there that you can't see.\r<BR>\n" );
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document.write( "the hole is because when you try to evaluate the function at x = 2, the answer is undefined.\r<BR>\n" );
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document.write( "it does, however, show you that, as you approach x = 2, the answer will approach x = 2/3.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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