document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1120561>Question 1120561</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the function below.<BR>\n" );
document.write( "lim x -> 3 (sqrt 1+x) = 2<BR>\n" );
document.write( "What values of x guarantee that f(x)=(sqrt 1+x) is within 0.0002 units of 2?<BR>\n" );
document.write( "If x is within _____ units of 3, then f(x) is within 0.0002 units of 2. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #736237 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=736237\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you want the answer to be within .0002 of 2.\r<BR>\n" );
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document.write( "this means the answer will be between 1.9998 and 2.0002.\r<BR>\n" );
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document.write( "solve the equation for each of those and you should have your limits.\r<BR>\n" );
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document.write( "sqrt(1+x) = 1.9998\r<BR>\n" );
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document.write( "square both sides to get 1+x = (1.9998)^2\r<BR>\n" );
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document.write( "subtract 1 from both sides to get x = (1.9998)^2 - 1\r<BR>\n" );
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document.write( "solve for x to get x = 2.99920004.\r<BR>\n" );
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document.write( "next:\r<BR>\n" );
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document.write( "sqrt(1+x) = 2.0002\r<BR>\n" );
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document.write( "square both sides to get 1+x = (2.0002)^2\r<BR>\n" );
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document.write( "subtract 1 from both sides to get x = 2.0002^2 - 1\r<BR>\n" );
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document.write( "solve for x to get x = 3.00080004.\r<BR>\n" );
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document.write( "if x is between 2.99920004 and 3.00080004, then sqrt)(1+x) will be within 1.9998 and 2.0002.\r<BR>\n" );
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document.write( "this can be seen graphically.\r<BR>\n" );
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document.write( "the first graph is a far out look.\r<BR>\n" );
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document.write( "the values of x are so close together, that you can't distinguish between the 3 vertical lines.\r<BR>\n" );
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document.write( "the second graph is a near end look.\r<BR>\n" );
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document.write( "now you can see that the values of x shown lead to the values of y shown.\r<BR>\n" );
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document.write( "here's the graphs.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/072803.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/072804.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "this could be modeled by an absolute value equation.\r<BR>\n" );
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document.write( "that equation would be:\r<BR>\n" );
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document.write( "|sqrt(1+x)-2| <= .0002\r<BR>\n" );
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document.write( "when the expression within the absolute value sign is positive, you get <BR>\n" );
document.write( "sqrt(1+x) - 2 <= .0002\r<BR>\n" );
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document.write( "add 2 to both sides of the equation and you get sqrt(1+x) <= 2.0002.\r<BR>\n" );
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document.write( "when the expression within the absolute value signs is negative, you get -(sqrt(1+x) - 2) <= .0002\r<BR>\n" );
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document.write( "multiply both sides of that inequality by -1 and you get sqrt(1+x) - 2 >= -.0002\r<BR>\n" );
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document.write( "multiplying both sides of an inequality by a negative number reverses the inequality.\r<BR>\n" );
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document.write( "add 2 to both sides of that inequality to get sqrt(1+x) >= 1.9998\r<BR>\n" );
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document.write( "either way you look at it, your answer will be the same.\r<BR>\n" );
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document.write( "when x is equal to 2.99920004, it is within 3 - 2.99920004 = .00079996 units of 3.\r<BR>\n" );
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document.write( "when x is equal to 3.00080004, it is within 3.0080004 - 3 = .00080004 units of 3.\r<BR>\n" );
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document.write( "the differences from 3 are not the same.\r<BR>\n" );
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document.write( "to guarantee one value that will put the result within .0002 of 2, you would need to pick the smaller of the two results.\r<BR>\n" );
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document.write( "that would be .00079996 units.\r<BR>\n" );
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document.write( "since your answer doesn't require you to round any intermediate or final results, i would go with that.\r<BR>\n" );
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document.write( "note that the answer shown in the calculator is rounded to 8 decimal digits.\r<BR>\n" );
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document.write( "i had to go to excel to get more decimal digits.\r<BR>\n" );
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document.write( "the answer given by excel is:\r<BR>\n" );
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document.write( "calculator shows .00079996 while excel shows 0.0007999599999997110<BR>\n" );
document.write( "calculator shows .00080004 while excel shows 0.0008000399999996690\r<BR>\n" );
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document.write( "the reason why the plus and minus units of x are not the same has to do with the squaring of both sides required to find the answer.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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