document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1122093>Question 1122093</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the following limit. lim x&#8594;3 (x^2 + 9) <BR>\n" );
document.write( "Find the limit L. \r<BR>\n" );
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document.write( "Well I know that the limit L is equal to 18, but then it asks me the following:\r<BR>\n" );
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document.write( "(a) Find &#948; > 0 such that |f(x) &#8722; L| < 0.01 whenever 0 < |x &#8722; c| < &#948;. (Round your answer to five decimal places.)\r<BR>\n" );
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document.write( "(b) Find &#948; > 0 such that |f(x) &#8722; L| < 0.005 whenever 0 < |x &#8722; c| < &#948;. (Round your answer to five decimal places.)\r<BR>\n" );
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document.write( "What is this trying to tell me to do? I am a very lost and would like to better understand. Thank you!  <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 #738172 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=738172\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> limit of x^2 + 9 is equal to 18 as x approaches 2.\r<BR>\n" );
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document.write( "you want to find a delta > 0 such that |x^2 + 9 - 18| < .01 when |x - 3| < delta\r<BR>\n" );
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document.write( "x^2 + 9 - 18 simplifies to x^2 - 9\r<BR>\n" );
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document.write( "therefore you want to find a delta > 0 such that |x^2 - 9| < .01 when |x - 3| < delta\r<BR>\n" );
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document.write( "|x^2 = 9| < .01 becomes -.01 < x^2 = 9 < .01 which becomes 8.99 < x^2 < 9.01\r<BR>\n" );
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document.write( "solve for x to get = sqrt(8.99) < x < sqrt(9.01)\r<BR>\n" );
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document.write( "subtract 3 from all sides to get sqrt(8.99) - 3 < x - 3 < sqrt(9.01) - 3\r<BR>\n" );
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document.write( "your delta will be either |sqrt(8.99) - 3| or it will be|sqrt(9.01) - 3|.\r<BR>\n" );
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document.write( "|sqrt(8.99) - 3| is equal to .0016671299\r<BR>\n" );
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document.write( "|sqrt(9.01) - 3| is equal to .001666204.\r<BR>\n" );
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document.write( "when |sqrt(8.99) - 3| < .0016671299, you get 2.99833287 < x < 3.00166713\r<BR>\n" );
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document.write( "when x = 2.99833287, you get |x^2 - 9| = .01 which is smaller than or equal to .01\r<BR>\n" );
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document.write( "when x = 3.00166713, you get |x^2 - 9| = .0100055586 which is not smaller than or equal to .01.\r<BR>\n" );
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document.write( "so, delta = .9915561288 will not work.\r<BR>\n" );
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document.write( "when |sqrt(9.01) - 3| is equal to .001666204, you get 2.998333796 < x < 3.001666204.\r<BR>\n" );
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document.write( "when x = 2.998333796, you get |x^2 - 9| = .0099944475 which is smaller than or equal to .01\r<BR>\n" );
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document.write( "when x = 3.001666204, you get |x^2 - 9| = .01 which is smaller than or equal to .01\r<BR>\n" );
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document.write( "so, delta = .001666204 will work.\r<BR>\n" );
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document.write( "as long as |x - 3| < .001666204, |x^2 - 9| will be smaller than .01.\r<BR>\n" );
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document.write( "your delta is .001666284 which rounds to .00167\r<BR>\n" );
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document.write( "use the same procedure for |x^2 - 9| < .005\r<BR>\n" );
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document.write( "|x^2 - 9| < .005 means -.005 < x^2 - 9 < .005\r<BR>\n" );
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document.write( "add 9 to all sides to get 8.995 < x^2 < 9.005\r<BR>\n" );
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document.write( "take the square root of all sides to get sqrt(8.995) < x < sqrt(9.005)\r<BR>\n" );
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document.write( "subtract 3 from all sides to get sqrt(8.995) - 3 < x - 3 < sqrt(9.005) - 3\r<BR>\n" );
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document.write( "your delta will be either |sqrt(8.995) - 3| or it will be |sqrt(9.005) - 3|\r<BR>\n" );
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document.write( "that says your delta will be either .000833449106 or it will be .008332176247\r<BR>\n" );
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document.write( "when delta = .000833449106, 2.999166551 < x < 3.000833449\r<BR>\n" );
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document.write( "when x = 2.999166551, |x^2 - 9| = .005 which is smaller than or equal to .005\r<BR>\n" );
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document.write( "when x = 3.000833449, |x^2 - 9| = .0050013893 which is not smaller than or equal to .005\r<BR>\n" );
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document.write( "so, delta = .000833449106 will not work.\r<BR>\n" );
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document.write( "when delta = .008332176247, 2.999166782 < x < 3.000833218\r<BR>\n" );
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document.write( "when x = 2.999166782, |x^2 - 9| = .0049986115 which is smaller than .005\r<BR>\n" );
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document.write( "when x = 3.000833218, |x^2 - 9| = .005 which is smaller than or equal to .005\r<BR>\n" );
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document.write( "so, delta = .008332176247 will work.\r<BR>\n" );
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document.write( "as long as |x - 3| < .008332176247, |x^2 - 9| will be < .005\r<BR>\n" );
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document.write( "your delta is .008332176247 which rounds to .00833\r<BR>\n" );
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document.write( "this can also be solved graphically.\r<BR>\n" );
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document.write( "the following videos discuss the concept.\r<BR>\n" );
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document.write( "not all explanations are simple to follow, but hopefully you will get the idea from these videos, which i chose because they were simpler to follow than others, at least for me.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.youtube.com/watch?v=ftAuCXNAvtE\" target = \"_blank\">https://www.youtube.com/watch?v=ftAuCXNAvtE</a>\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.youtube.com/watch?v=NaufUAAKdgk\" target = \"_blank\">https://www.youtube.com/watch?v=NaufUAAKdgk</a>\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.youtube.com/watch?v=MAMPkdzsAas\" target = \"_blank\">https://www.youtube.com/watch?v=MAMPkdzsAas</a>\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.youtube.com/watch?v=gE6n3Da0U4k\" target = \"_blank\">https://www.youtube.com/watch?v=gE6n3Da0U4k</a>\r<BR>\n" );
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document.write( "in some cases, the deltas will be the same, in which case it's a no brainer what the delta needs to be.\r<BR>\n" );
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document.write( "in other cases, as in your problem, the deltas will be different which means you have to determine which of the deltas is the one that will satisfy the requirements of |f(x) - L| < epsilon.\r<BR>\n" );
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