Section 2.6: Limits at Infinity; Horizontal Asymptotes
Definition: If the values of f (x) can be made arbitrarily close to L by taking x sufficiently
large, then we say that the limit of f (x), as x approaches ∞, equals L. This is denoted by
lim f (x) = L.
x→∞
Similarly, if the values of f (x) can be made arbitrarily close to L by taking x sufficiently
large negative, then we say that the limit of f (x), as x approaches negative −∞, equals L.
This is denoted by
lim f (x) = L.
x→−∞
The line y = L is called a horizontal asymptote of y = f (x) if either
lim f (x) = L
x→∞
or
Example: Evaluate the following limits.
(a) lim
x→∞
1
x
1
x→−∞ x3
(b) lim
1
lim f (x) = L.
x→−∞
(c) lim (2x − x2 )
x→∞
(d) lim (x − 3x3 )
x→−∞
7x3 + 4x
x→∞ 2x3 − x2 + 3
(e) lim
2x2 + 5x − 6
x→∞ 4x3 − 2x2 + 1
(f) lim
x3 − 2x + 3
x→∞
4 − 3x2
(g) lim
2
√
2x2 + 1
(h) lim
x→∞ 3x − 5
√
(i) lim
x→−∞
x2 + 4x
4x + 1
√
(j) lim ( x2 + 3x + 1 − x)
x→∞
3
Example: Find all vertical and horizontal asymptotes of the following functions.
(a) f (x) =
(b) f (x) =
x2 + 4
x2 − 1
x2
x+3
+ 8x + 15
4