# Exponential of a rational function.

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The rational function is defined over , so is .

We compute four limits:

It follows that the lines whose respective equations are and are asymptotes to the graph of . Pay attention that the existence of the vertical asymptote is shown by the limit on the right at 2, but that on the left at 2 the function has a finite limit.

As a composition of two differentiable functions, is differentiable over .

thus all over . It follows that is a decreasing function on and on .

The graph and its asymptotes are displayed in Figure 5.

Noah Dana-Picard 2007-12-28