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    <blockquote> <p>Name a function that is neither O(n) nor Omega(n)</p> </blockquote> <p>Saying <code>f ∈ O(g)</code> means the quotient</p> <pre><code>f(x)/g(x) </code></pre> <p>is bounded from above for all sufficiently large <code>x</code>.</p> <p><code>f ∈ Ω(g)</code> on the other hand means the quotient</p> <pre><code>f(x)/g(x) </code></pre> <p>is bounded below away from zero for all sufficiently large <code>x</code>.</p> <p>So to find a function that is neither <code>O(n)</code> nor <code>Ω(n)</code> means finding a function <code>f</code> such that the quotient</p> <pre><code>f(x)/x </code></pre> <p>becomes arbitrarily large, and arbitrarily close to zero on every interval <code>[y, ∞)</code>.</p> <blockquote> <p>I'm thinking this may be a correct answer: <code>(n^3 (1 + sin n))</code> is neither O(n) nor Omega(n).</p> </blockquote> <p>Let's plug it in our quotient:</p> <pre><code>(n^3*(1 + sin n))/n = n^2*(1 + sin n) </code></pre> <p>The <code>n^2</code> grows to infinity, and the factor <code>1 + sin n</code> is larger than 1 for roughly three out of every six <code>n</code>. So one every interval <code>[y, ∞)</code> the quotient becomes arbitrarily large. Given an arbitrary <code>K &gt; 0</code>, let <code>N_0 = y + K + 1</code> and <code>N_1</code> the smallest of <code>N_0 + i, i = 0, 1, ..., 4</code> such that <code>sin (N_0+i) &gt; 0</code>. Then <code>f(N_1)/N_1 &gt; (y + K + 1)² &gt; K² + K &gt; K</code>.</p> <p>For the <code>Ω(n)</code> part, it's not so easy to prove, although I believe it is satisfied.</p> <p>But, we can modify the function a bit, retaining the idea of multiplying a growing function with an oscillating one in such a way that the proof becomes simple.</p> <p>Instead of <code>sin n</code>, let us choose <code>cos (π*n)</code>, and, to offset the zeros, add a fast decreasing function to it.</p> <pre><code>f'(n) = n^3*(1 + cos (π*n) + 1/n^4) </code></pre> <p>now,</p> <pre><code> / n^3*(2 + 1/n^4), if n is even f'(n) = &lt; \ 1/n , if n is odd </code></pre> <p>and it is obvious that <code>f'</code> is neither bounded from above, nor from below by any positive constant multiple of <code>n</code>.</p>
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    1. POAlgorithm Analysis (Big O and Big Omega)
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    1. USDaniel Fischer
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