Math 175
Notes and Learning Goals
Lesson 5-1a
End Behavior
• The end behavior of a function describes what the tail ends of the graph do as x goes to ±∞.
• Limits are a tool that can be used to describe or find the end behavior: lim f (x)
x→∞
• There are three basic types of end behavior: infinite, finite or non-existent.
• The speed two functions run away to infinity can be compared by the limit of their ratio:
Given that both lim f (x) = ∞ and lim g(x) = ∞ then
x→∞
x→∞
f (x)
= 0 then f (x) goes to ∞ slower than g(x): f (x) g(x)
x→∞ g(x)
f (x)
= ±∞ then f (x) goes to ∞ faster than g(x): f (x) g(x)
– If lim
x→∞ g(x)
f (x)
– If lim
6= 0 is finite then f (x) goes to ∞ about the same speed as g(x): f (x) ≈ g(x)
x→∞ g(x)
– If lim
• f (x) g(x) says that for large values of x, f (x) is sufficiently smaller than g(x).
• Examples:
5x2
5
=
lim
= 0 so 5x2 goes to infinity slower than 7x3 : 5x2 7x3
x→∞ 7x3
x→∞ 7x
lim
5
5x2
5
= lim = so 5x2 goes to infinity about the same speed as 3x2 : 5x2 ≈ 3x2
2
x→∞ 3x
x→∞ 3
3
lim
• The order of elementary function families ordered from slowest to fastest:
Logarithms
loga (x)
Roots
x1/n
Powers
xn
Exponential
ax
• Each elementary family is subdivided further: x2 is slower than x5 .
• All logarithms approach infinity about the same speed, but are eventually slower than any root.
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Limits at Infinity
• When finding the limit at infinity only the fastest terms are important.
• When working with a ratio, determine the limit by comparing the ratio of the dominate terms.
4x2 − 5x + 1
4x2
=
lim
= −4
x→∞ x − x2 + 2
x→∞ −x2
Example: lim
• This can be formalized by factoring out the dominate terms in the numerator and denominator:
5x
1
4x2 1 − 4x
4x2 (1 − 0 + 0)
4x2 − 5x + 1
2 + 4x2
Example: lim
=
lim
= −4
=
lim
x→∞ −x2 (0 + 1 − 0)
x→∞ x − x2 + 2
x→∞ −x2 − x2 + 1 − 22
x
x
In each of the fractions the fastest term was in the denominator pulling the ratio to zero.
• When looking at end behavior, limx→∞ f (x), using the above method to look at the ratios making
most the terms zero is highly effective. The conclusion of this is that in practice ignore all except
the dominate terms as x → ∞.
• When unsure if one term is faster (or slower) than another, graph the ratio
determine if the tail end goes to 0, ∞, or some other non-zero limit.
f (x)
g(x)
(or
g(x)
)
f (x)
and
• Be able to calculate the limit as x → ∞ of a ratio by comparing the dominate terms.
• If the limit as x → ∞ is finite, then we say that f (x) has a horizontal asymptote at the limit.
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