document.write( "Question 361025: prove: if t>0, there exists b such that if x>b then (1/(sqrt(x))) < t \n" ); document.write( "

Algebra.Com's Answer #257569 by robertb(5830) $\"\"$ $\"About$

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Let $\"b+=+1%2Ft%5E2\"$ .
\n" ); document.write( "Proof:\r
\n" ); document.write( "\n" ); document.write( "If $\"b+=+1%2Ft%5E2\"$ , then because b < x, $\"1%2Ft%5E2%3Cx\"$ .
\n" ); document.write( " $\"1%3Cxt%5E2\"$ , since t>0.
\n" ); document.write( " $\"0%3Ct%5E2x-1+=+%28t%2Asqrt%28x%29-1%29%28t%2Asqrt%28x%29%2B1%29\"$ , after transposition and factoring the quadratic expression on the right.
\n" ); document.write( " $\"0%3Ct%2Asqrt%28x%29-1\"$ , after dividing both sides by $\"t%2Asqrt%28x%29%2B1\"$ , which is surely positive. So
\n" ); document.write( " $\"1%3Ct%2Asqrt%28x%29\"$ ,and thus
\n" ); document.write( " $\"1%2Fsqrt%28x%29%3Ct\"$ , since $\"sqrt%28x%29\"$ is positive. \n" ); document.write( "

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