Download Lecture Notes – MTH 251 2.5. Limits at Infinity We shall contrast

Lecture Notes – MTH 251
2.5. Limits at Infinity
We shall contrast Infinite Limits with Limits at Infinity. Whereas Infinite Limits describe
when the value of a function (dependent variable) grows without bound, limits at infinity are
used to describe behavior of functions as the independent variable grows without bound.
On graphs, you may see these as horizontal asymptotes:
If limx→∞ f (x) = L, then f has a horizontal asymptote at y = L.
Note: To stay true to the definition of functions, notice that functions cannot cross vertical
asymptotes (at least not in a continuous sense), but they can cross through (or even coincide
with) horizontal asymptotes.
1
2.5. LIMITS AT INFINITY
Definition 2.5.1
We can say:
lim f (x) = L
x→+∞
means “We can make f (x) as close as desired to L by letting x be as large (positive) as
needed.”
y
L
x
Other ideas we can consider:
(1) The behavior of f as x → −∞.
(2) Infinite limits at infinity:
lim f (x) = +∞,
x→+∞
(3) Limit laws and the Squeeze Theorem, like those presented in section 2.3, also apply in to limits at infinity.
When it comes to computing limits at infinity, it is often helpful to recall these particular
cases:
2
2.5. LIMITS AT INFINITY
Example 2.5.2
Limit at infinity of a rational function:
(2x2 + x − 2) x−2
x→∞ (x2 − 2x − 3) x−2
2x2 + x − 2
=
x→∞ x2 − 2x − 3
lim
lim
2+
x→∞ 1 −
=
lim
1
x
2
x
−
−
limx→∞ 2 +
=
limx→∞ 1 −
2
x2
3
x2
1
x
2
x
−
−
2
x2 3
x2
2+0−0
1−0−0
=
= 2.
We can also consider the limit at −∞:
limx→−∞ 2 + limx→−∞
2x2 + x − 1
=
lim 2
x→−∞ x − 2x − 3
limx→−∞ 1 − limx→−∞
1
x
2
x
− limx→−∞
− limx→−∞
= 2.
If we plot y =
2x2 +x−2
,
x2 −2x−3
we see:
y
4
f
3
2
1
−15
−10
−5
5
−1
−2
−3
−4
3
10
15
x
2
x2
3
x2
2.5. LIMITS AT INFINITY
Notice the trick here is to multiply top and bottom by a particular power function that gives
terms in the numerator and denominator for which the limits are not infinity. This is a
common technique.
After evaluating limits of rational functions at infinity, the following patterns will emerge:
Proposition 2.5.3: Limits at infinity of rational functions
(1) If p and q are polynomial functions of the same degree:
p (x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 ,
q (x) = bn xn + bn−1 xn−1 + · · · + b2 x2 + b1 x + b0 ,
in particular, an and bn are nonzero, then:
p (x)
an
an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0
= .
= lim
n
n−1
2
x→±∞ q (x)
x→±∞ bn x + bn−1 x
+ · · · + b2 x + b1 x + b0
bn
lim
(2) If the degree of p exceeds the degree of q:
p (x) = am xm + am−1 xm−1 + · · · + a2 x2 + a1 x + a0 ,
q (x) = bn xn + bn−1 xn−1 + · · · + b2 x2 + b1 x + b0 ,
where am 6= 0, bn 6= 0, and m > n, then:
p (x)
am xm + am−1 xm−1 + · · · + a2 x2 + a1 x + a0
= lim
= +∞,
x→±∞ q (x)
x→±∞ bn xn + bn−1 xn−1 + · · · + b2 x2 + b1 x + b0
lim
if am and bn have the same sign, and −∞ if they have different sign.
(3) If the degree of q exceeds the degree of p, that is, m < n, then:
am xm + am−1 xm−1 + · · · + a2 x2 + a1 x + a0
p (x)
= lim
= 0.
x→±∞ bn xn + bn−1 xn−1 + · · · + b2 x2 + b1 x + b0
x→±∞ q (x)
lim
4
2.5. LIMITS AT INFINITY
Example 2.5.4
Limit at infinity of a not-so-rational function:
√
3
x6 + 8
√
lim
=
x→∞ 4x2 +
3x4 + 1
=
=
=
=
=
=
√
3
x6 + 8 · x−2
√
x→∞ 4x2 +
3x4 + 1 · x−2
p
3
(x6 + 8) · x−6
p
lim
x→∞ 4 +
(3x4 + 1) · x−4
q
3
1 + x86
q
lim
x→∞
4 + 3 + x14
q
limx→∞ 3 1 + x86
q
1
limx→∞ 4 + 3 + x4
q
3
limx→∞ 1 + limx→∞ x86
q
limx→∞ 4 + limx→∞ 3 + limx→∞
√
3
1+0
√
4+ 3+0
1
√ .
4+ 3
lim
y
3.0
f
2.5
2.0
1.5
1.0
0.5
1
A quick use of a calculator reveals that
2
3
1√
4+ 3
≈ 0.174.
5
4
x
1
x4
2.5. LIMITS AT INFINITY
Example 2.5.5
Evaluate:
16x + 7
√
,
x→+∞ 6x +
4x2 + 1
16x + 7
√
.
x→−∞ 6x +
4x2 + 1
lim
lim
Solution. For the first limit, we shall multiply the top and bottom by x1 , and use the
q
fact that x1 = x12 , for x > 0. This is okay, as we are letting x → +∞:
16 + 7/x
16x + 7
√
√
= lim
x→∞ 6x +
4x2 + 1 x→∞ 6 + 4x2 + 1 · 1/x
lim
= lim
x→∞
= lim
x→∞
6+
√
16 + 7/x
4x2 + 1 ·
p
1/x2
16 + 7/x
6+
p
(4x2 + 1) · 1/x2
16 + 7/x
p
x→∞ 6 +
4 + 1/x2
= lim
=
16
√
6+ 4
= 2.
It may seem conceivable that the limit would be the same if we let x → −∞. However,
if we graph the operand:
y
6
5
4
3
2
1
f
−8 −7 −6 −5 −4 −3 −2 −−11
−2
−3
−4
−5
−6
1 2 3 4 5 6 7 8
6
x
2.5. LIMITS AT INFINITY
the limit letting x → −∞ appears different. This is because when x < 0.
p
− x1 , oweing to the fact that y 2 = |y|, in general, and not simply y:
q
1
x2
= x1 =
16x + 7
16 + 7/x
√
√
= lim
x→−∞ 6x +
4x2 + 1 x→−∞ 6 + 4x2 + 1 · 1/x
lim
= lim
x→−∞
6−
√
16 + 7/x
4x2 + 1 ·
p
1/x2
16 + 7/x
p
x→−∞ 6 −
(4x2 + 1) · 1/x2
= lim
16 + 7/x
p
x→−∞ 6 −
4 + 1/x2
= lim
=
16
√
6− 4
= 4.
7
h
2.5. LIMITS AT INFINITY
Proposition 2.5.6: Special limits at infinity
(1) limx→±∞
1
xn
= 0 whenever n is positive,
(2) limx→+∞ p (x) = +∞ whenever p is a nonconstant polynomial with positive lead
coefficient,
(3) limx→−∞ p (x) = +∞ or −∞ whenever p is a nonconstant polynomial with
positive lead coefficient, depending on whether p has even degree or odd degree:
(4)
(5)
(6)
(7)
−∞ if odd degree,
 +∞ if even degree.


+∞ if r > 0,




limt→+∞ ert = 1
if r = 0,





0
if r < 0,



0
if r > 0,




limt→−∞ ert = 1
if r = 0,





+∞ if r < 0,



+∞ if b > 1,




limt→+∞ bt = 1
if b = 1,





0
if 0 < b < 1,



0
if b > 1,




limt→−∞ bt = 1
if b = 1,





+∞ if 0 < b < 1,
As a consequence of the previous two limits, we also have the following infinite limits:
(8) limt→+∞ logb t = +∞, if b > 1 (includes natural logarithm),
(9) limt→0+ logb t = −∞, if b < 1.
8
2.5. LIMITS AT INFINITY
Natural Exponential function as a limit/Discrete vs. Continuous compounding interest. Recall interest rate problems. When you make an initial deposit P0 into a
bank account with annual interest rate r, the function P that gives the balance as a function
of time depends on how often interest is compounded, or if it is continuously compounded.
If interest is compounded n times annually, then
r bntc
P (t) = P0 1 +
.
n
The notation bntc indicates the floor function, and rounds nt down to the nearest integer.
This is done since in actuality, the balance does not continuously rise, but rather jumps up
every
1
n
years.
If interest is compounded continuously, you were given the following formula:
P (t) = P0 ert ,
where e was some constant that is approximately equal to 2.71.
As it turns out,
lim
n→∞
r bntc
1+
= ert
n
for each real number r and t. This comes from one definition of the exponential function:
Definition 2.5.7: Natural Exponential Function
We define the natural exponential function as follows:
ex = lim
n→∞
1+
x n
.
n
As this is what makes it possible to define exponentiation by an arbitrary real number, it
must be understood that this limit is taken with over positive integers n only (n = 1, 2, 3, . . . ).
9
2.5. LIMITS AT INFINITY
Since we define ex as a limit, we must be careful to make sure the limit always exists. As it
turns out,
Theorem 2.5.8
For all real numbers x,
lim
n→∞
x n
1+
n
exists. Moreover, this limit is always positive.
Proof that this limit exists is beyod the scope of this class.
Furthermore, the natural exponential function enjoys the properties of exponentiation we
are already familiar with for integer exponents:
Theorem 2.5.9
For all real numbers x and y,
e0 = 1,
ex+y = ex · ey ,
ex−y =
ex
.
ey
Again, the proof is beyond the scope of this class.
The exponential function also satisfies the following limits:
Theorem 2.5.10
lim ex = ec , c ∈ R,
x→c
lim ex = 0,
x→−∞
10
lim ex = ∞.
x→∞
2.5. LIMITS AT INFINITY
Example 2.5.11
Limit at infinity involving exponential functions:
(ex + 1) e−x
x→∞ (ex − 1) e−x
ex + 1
=
x→∞ ex − 1
lim
lim
1 + e−x
x→∞ 1 − e−x
1+0
=
1−0
=
lim
= 1
On the other hand:
0+1
ex + 1
=
x
x→−∞ e − 1
0−1
lim
= −1
Notice in the second case that we did not need to multiply the numerator and denominator by anything because ex → 0 as x → −∞. Also notice that these two limits are
different.
Here is our graph to visualize our limits:
y
3
f
2
1
−4
−3
−2
−1
1
−1
−2
−3
11
2
3
4
x