document.write( "Question 1120562: Consider the function below.
\n" ); document.write( "lim x -> 3 x^2=9
\n" ); document.write( "What values of x guarantee that f(x) = x^2 is within 0.2 units of 9?
\n" ); document.write( "If x is within _____ units of 3, then f(x) is within 0.2 units of 9. \n" ); document.write( "

Algebra.Com's Answer #736235 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "You want to have, as the problem requires\r\n" );
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document.write( "    |x^2 -9|  <=  0.2.\r\n" );
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document.write( "It means\r\n" );
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document.write( "    -0.2 <= x^2 - 9 <= 0.2\r\n" );
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document.write( "    9-0.2 <= x^2 <= 9+0.2\r\n" );
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document.write( "    8.8   <= x^2 <= 9.2.      Now take the square root from both sides\r\n" );
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document.write( "     <= x <=        (1)\r\n" );
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document.write( "Notice that   = 2.966. . .   and   = 3.033.\r\n" );
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document.write( "Therefore,  inequality (1)  implies\r\n" );
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document.write( "     2.966 <= x <= 3.033    (with 3 decimals after the decimal point).\r\n" );
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document.write( "It means that  |x - 3|  <= 0.033.\r\n" );
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document.write( "Answer.  If x is within  0.033  units of 3, then f(x) is within 0.2 units of 9.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Notice that $\"sqrt%280.2%29\"$ = 0.447 (approximately.\r
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\n" ); document.write( "\n" ); document.write( "Therefore, it is NOT ENOUGH to have |x-3| <= sqrt(0.2) in order for the required inequality was in place.\r
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\n" ); document.write( "\n" ); document.write( "It shows that the solution by the other tutor is incorrect.\r
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