Generalizations.

The notion of an asymptote can be generalized to any curve

Definition 4.9.1   Let $ f$ be a function defined on a neighborhood of $ +\infty$ and let $ C$ be the curve whose equation is $ y=g(x)$ , where $ g$ is a function defined on a neighborhood of $ +\infty$ . The curve $ C$ is an asymptote to the graph of $ f$ if $ \underset{x \rightarrow +\infty}{\lim} [f(x)-g(x)]=0$ .

Example 4.9.2   The parabola whose equation is $ y=x^2$ is an asymptote to the curve $ \mathcal{C}$ named Neptune's trident and whose equation is $ y=x^2+\frac 1x$ (see Figure 17)

Figure 17: Neptune's trident.
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Noah Dana-Picard 2007-12-28