SOLUTION: prove: if t>0, there exists b such that if x>b then (1/(sqrt(x))) < t

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Question 361025: prove: if t>0, there exists b such that if x>b then (1/(sqrt(x))) < t
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Let b+=+1%2Ft%5E2.
Proof:
Ifb+=+1%2Ft%5E2, then because b < x, 1%2Ft%5E2%3Cx.
1%3Cxt%5E2, since t>0.
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.
0%3Ct%2Asqrt%28x%29-1, after dividing both sides by t%2Asqrt%28x%29%2B1, which is surely positive. So
1%3Ct%2Asqrt%28x%29,and thus
1%2Fsqrt%28x%29%3Ct, since sqrt%28x%29is positive.