Calculus AB and Calculus BC

CHAPTER 2 Limits and Continuity

B. ASYMPTOTES

Horizontal asymptote

The line y = b is a horizontal asymptote of the graph of y = f (x) if

Image

The graph of Image (Figure N2–4) has the x-axis (y = 0) as the horizontal asymptote.

So does the graph of Image (Figure N2–5).

The graph of Image has the line y = 1 as the horizontal asymptote, as shown at the right.

Image

Vertical asymptote

The line x = a is a vertical asymptote of the graph of y = f (x) if one or more of the following holds:

Image

The graph of Image (Figure N2–4) has x = 0 (the y-axis) as the vertical asymptote.

The graph of Image (Figure N2–5) has x = 1 as the vertical asymptote.

The graph of Image (Figure N2–6) has the line x = 2 as the vertical asymptote.

EXAMPLE 8

From the graph of Image in Figure N2–7, describe the asymptotes of k using limits.

SOLUTION: We see that y = 2 is a horizontal asymptote, since

Image

Also, x = 3 is a vertical asymptote; the graph shows that

Image

Image

FIGURE N2–7