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For horizontal asymptotes,
3x
f 1x2
x 2 x 6
3x
x 2 a 1 1 x 6 x 2 b
(Factor)
(Simplify)
lim f 1x2 lim
xS q xSq
3
x a 1 1 x 6 x 2 b
3
x 0
Similarly, we can show lim f 1x2 0. Therefore, y 0 is a horizontal asymptote
xS q
of the graph of f 1x2 for both large positive and negative values of x.
As x becomes large positively, f 1x2 7 0, so the graph is above the horizontal
asymptote. As x becomes large negatively, f 1x2 6 0, so the graph is below the
horizontal asymptote.
This diagram illustrates the behaviour of the graph as it nears the asymptotes:
y
3
2
1
x = –2
x
–2 –1 0 1 2 3 4
–1
y = 0
–2
x = 3
–3
Horizontal Asymptotes and Limits at Infinity
If lim f 1x2 L or lim f 1x2 L, we say that the line y L is a horizontal
xS q
xS q
asymptote of the graph of f 1x2.
NEL
CHAPTER 4 189