title: Complexity - determining the order of growth
body: <p>I understand how to calculate a function's complexity for the most part. The same goes for determining the order of growth for a mathematical function. [I probably don't understand it as much as I think I do, which is why I'm probably asking this.] For instance:</p>
  
  <p><code>an^3 + bn^2 + cn + d</code> could be written as O(n^3) in big-o notation, since for large enough <code>n</code>the values of the term <code>bn^2 + cn + d</code> are insignificant in comparison to <code>an^3</code> (the constant coefficients a, b, c and d are left out as well, as their contribution to the value become insignificant, too).</p>
  
  <p>What I don't understand is, how does this work when the leading term is involved in some sort of division? For instance:</p>
  
  <p><code>a/n^3 + bn^2</code> or <code>n^3/a + bn^2</code>  </p>
  
  <p>Let n=100, a=1000 and b=10 for the former formula, then we have</p>
  
  <p><code>n^3/a = 100^3/1000 = 1000</code> and <code>bn^2 = 10*100^2 = 100,000</code></p>
  
  <p>or even more dramatic for the latter - in this case the leading term is not only growing slowly as above, but it's also shrinking, isn't it?:</p>
  
  <p><code>a/n^3 = 1000/100^3 = 0.001</code> and <code>bn^2 = 100,000</code> as above.</p>
  
  <p>In both cases the second term contributes much more, so isn't it <code>n^2</code>that actually determines the order of growth?</p>
  
  <p>It gets even more complicated (for me, at least) when the leading term is followed by a subtraction (<code>a/n^3 - bn^2</code>) or when the second term is also a division (<code>n^3/a + n^2/b)</code> or when both are divisions but in mixed order (<code>a/n^3 + n^2/b</code>), etc.</p>
  
  <p>The list seems endless, so my general question is, how to understand and handle formulas that involve division (and subtraction) in order to determine the order of growth for a given function?</p>