Asymptotes: Definitions and Examples

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ASYMPTOTES. By Joanna Gutt-Lehr, Pinnacle Learning Lab, last updated 12/ 2009. Let y = f(x). Vertical asymptote: x = c. As x → c, y → ± ∞ or. The value x = c is ...
ASYMPTOTES Let y = f(x). Vertical asymptote: x = c As x → c, y → ± ∞ or The value x = c is not in the domain of the function f. The graph of f cannot intersect its vertical asymptote.

Horizontal asymptote: y = b As x → ± ∞, y → b or The graph of f can intersect its horizontal asymptote.

Slant asymptote: y = ax + b As x → ± ∞, f(x) → y = ax + b, a ≠ 0 or The graph of f can intersect its horizontal asymptote.

Examples: y

y

2

-2

2

4

x

5

-2 x

Vertical asymptote x = 5

Horizontal asymptote (two-sided) y=2

Horizontal asymptote (one-sided) y=3

y

40 30 4

20 10

3

-50 -40 -30 -20 -10

10

20

30

40

2

50

-10

-5

5

-20

x

-30

x

=5

Vertical asymptotes: x = -5, x = 5 Horizontal asymptote: y = 3

-4

-2

2

-40

Vertical Asymptote x = 0 Slant asymptote (two-sided) y= x–

By Joanna Gutt-Lehr, Pinnacle Learning Lab, last updated 12/2009

Slant asymptote (one-sided) y=x+2