As promised in Example 7 of this section, we will give you
Chapter 1, Problem 1.162(choose chapter or problem)
As promised in Example 7 of this section, we will give you a chance to prove algebraically that \(p(x)=x /\left(1+x^{2}\right)\) is bounded.
(a) Graph the function and find the smallest integer k that appears to be an upper bound.
(b) Verify that \(x /\left(1+x^{2}\right)<k\) by proving the equivalent inequality \(k x^{2}-x+k>0\). (Use the quadratic formula to show that the quadratic has no real zeros.)
(c) From the graph, find the greatest integer k that appears to be a lower bound.
(d) Verify that \(x /\left(1+x^{2}\right)>k\) by proving the equivalent inequality \(k x^{2}-x+k<0\).
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