Rational function of an exponential.

.

This function is defined on .

The function is the composition of a rational function on the exponential; thus, it is differentiable on its domain. We have:

This derivative is never equal to 0, and for any . Hence, the function decreases on and on .

Let us compute the limits of at the open ends of the domain:

Therefore, the line whose equation is is an asymptote to the graph of .

As the exponential function with basis has as its limit at , we need some lagberaic work. We have:

Now we have:

Therefore, the line whose equation is is an asymptote to the graph of .

At , we need to compute one-sided limits:

The line whose equation is is an asymptote to the graph of .

Let us now compute the second derivative of :

It follows that

Therefore, is concave over and convex over ; it has no point of inflexion.

We display the graph of in Figure 4.

Noah Dana-Picard 2007-12-28