Bounds of the Reciprocal of a Quadratic

The reciprocal of a quadratic, if the quadratic is never zero, is bounded. We can find the bounds by completing the square for the quadratic.
Suppose we want to find the bounds of  
Numerator and denominator are both positive, and as  
\[x \rightarrow \infty 0\]
, the denominator tends to infinity also, so the fraction tends to zero. Hence  
\[0 \lt \frac{2}{x^2+4x+9}\]
Completing the square for the denominator gives  
To maximise this fraction we must minimise the denominator, which is the sum of non negative terms. The denominator is minimised when  
\[(x+2)^2=0 \rightarrow x=-2\]
For this value of  
  the fraction is equal to  
\[0 \lt \frac{2}{(-2+2)^2+5}=\frac{2}{0+5}=\frac{2}{5}\]
\[0 \lt \frac{2}{x^2+4x+9} \lt \frac{2}{5}\]
The graph of the function is shown below.

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