1. Recall 2.2 Infinite Limits and Vertical Asymptotes
Definition 1.1. The limit of f (x), as x approaches a from the right, is infinite
means that as x gets arbitrarily close to the value a, the value of f (x) gets arbitrarily
large. This is also written
lim f (x) = ∞
x→a
If the value of |f (x)| gets arbitrarily large, but f (x) < 0, for x close to a, then we
write
lim f (x) = −∞
x→a
Definition 1.2. If lim f (x) = ±∞, lim− f (x) = ±∞, or lim+ f (x) = ±∞, then the
x→a
x→a
x→a
vertical line x = a is a vertical asymptote of the curve y = f (x).
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2.6 Infinite Limits and Limits to Infinity
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2. Limits at infinity and Horizontal Asymptotes
Definition 2.1. We say the limit as x approaches infinity is L, written lim f (x) = L,
x→∞
if for some x large enough the graph of y = f (x) moves closer and closer to the line
y = L as one moves to the right. Moreover, in this case the graph y = f (x) has a
horizontal asymptote y = L.
Definition 2.2. We say the limit as x approaches negative infinity is L, written
lim f (x) = L, if for some x far enough to the left the graph of y = f (x) moves
x→−∞
closer and closer to the line y = L as one moves further to the left. Moreover, in this
case the graph y = f (x) has a horizontal asymptote y = L.
Example 2.1. The graph of y = f (x) is
y
1
2
x
1
(a) lim f (x) =
-1
x→−∞
(b) lim f (x) =
-2
x→∞
Example 2.2. Sketch a graph of a function satisfying the following conditions:
The domain for f is all real numbers except 2, f is continuous on its domain,
lim f (x) = ∞, lim f (x) = −∞, lim f (x) = 1, and f (−2) = 0.
x→2
x→∞
x→−∞
2.6 Infinite Limits and Limits to Infinity
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3. Some Special Limits that must be memorized
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1
and lim
x→∞ x
x→−∞ x
Example 3.1. Find the limits lim
Example 3.2. Find the limits lim sin x and lim cos x
x→∞
x→∞
Example 3.3. Find the limits lim arctan x and lim arctan x
x→∞
x→−∞
Example 3.4. Find the limits lim ex and lim ex
x→∞
x→−∞
4. Methods for special types of functions
Example 4.1. (Polynomials) Find the limits lim (−2x3 + 4x2 − x + 5) and lim (−2x3 + 4x2 − x + 5)
x→∞
x→−∞
90t2 + 4t + 5
t→−∞ 3t3 + t + 4
Example 4.2. (Rational Functions) Find the limit lim
3t3 + t + 4
Example 4.3. Find the limit lim
t→∞ 90t2 + 4t + 5
2.6 Infinite Limits and Limits to Infinity
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√
√
9x6 − 16
9x6 − 16
and
lim
Example 4.4. (Radicals) Find the limits lim
x→+∞
x→−∞
x3 + 1
x3 + 1
Example 4.5. (Radicals in a sum or difference) Find the limits lim (x +
x→−∞
√
and lim (x + x2 + 16x)
√
x2 + 16x)
x→+∞
Example 4.6. (Compositions) Find the limits lim arctan(x2 −x3 ) and lim arctan(x2 −
x→−∞
x→+∞
x3 )
Example 4.7. Sketch the graph of y =
and intercepts.
x2 − 1
. Include domain, asymptotes
x2 + 3x + 2