## 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

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

So does the graph of (Figure N2–5).

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

**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:

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

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

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

**EXAMPLE 8**

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

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

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

**FIGURE N2–7**