Asymptotes

The function f(x) = 1/x whne x approaches ± ∞ gets closer and closer to the x-axis, which is the line y = 0. The line y = 0 is called the asymptote of the graph.

Definition 4.2.1. The line y = L is a horizontal asymptote of the graph of f when

$$\underset{x\to -\mathrm{\infty }}{lim}f(x)=L\text{or}\underset{x\to \mathrm{\infty }}{lim}f(x)=L$$

Note that from this definition, it follows that the graph of a function of x can have at most two horizontal asymptotes − one to the right and one to the left. For rational functions the two asympotetes are equal. Function that are not rational, however, may approach different horizontal asymptotes to the right and to the left.

Example 4.2.2. Find each limit

$$\underset{x\to \mathrm{\infty }}{lim}\frac{3x-2}{\sqrt{2x^2+1}},\underset{x\to -\mathrm{\infty }}{lim}\frac{3x-2}{\sqrt{2x^2+1}}$$

it is easy calculated that the right limit is 3/2 and the left one is −3/2.  ■

Definition 4.2.4. The function f has a vertical asymptote at x₀ if

$$\underset{x\to x_0^{-}}{lim}f(x)=\pm \mathrm{\infty }\text{or}\underset{x\to x_0^{+}}{lim}f(x)=\pm \mathrm{\infty }$$

Definition 4.2.5. The function f has an oblique asymptote y = ax + b at ±∞ if

$$\underset{x\to \pm \mathrm{\infty }}{lim}[f(x)-(mx+q)]=0\text{\hspace{1em}4.1.6}$$

Eq 4.1.6 is equivalente to the followins pair of conditions

1. lim_{x ⟶ ± ∞} f(x)/x = m

2. lim_{x ⟶ ± ∞} [f(x) − mx] = q

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