\(\dfrac{2010}{2009}-\dfrac{2009}{2008}+\dfrac{1}{2008\times2009}\)
\(=\dfrac{2010\times2008}{2008\times2009}-\dfrac{2009\times2009}{2008\times2009}+\dfrac{1}{2008\times2009}\)
\(=\dfrac{2010\times2008-2009\times2009+1}{2008\times2009}\)
\(=\dfrac{\left(2009+1\right)\times2008-2009\times2009+1}{2008\times2009}\)
\(=\dfrac{2009\times2008+2008+1-2009\times2009}{2008\times2009}\)
\(=\dfrac{2009\times2008+2009-2009\times2009}{2008\times2009}\)
\(=\dfrac{2009\times\left(2008+1-2009\right)}{2008\times2009}\)
\(=\dfrac{2009\times0}{2008\times2009}\)
\(=0\)
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