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  
\[\frac{2}{x^2+4x+9}\]
.
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  
\[\frac{2}{x^2+4x+9}=\frac{2}{(x+2)^2+5}\]
.
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  
\[x\]
  the fraction is equal to  
\[0 \lt \frac{2}{(-2+2)^2+5}=\frac{2}{0+5}=\frac{2}{5}\]
.
Hence  
\[0 \lt \frac{2}{x^2+4x+9} \lt \frac{2}{5}\]
.
The graph of the function is shown below.