Course: Accelerated Engineering Calculus I
Instructor: Michael Medvinsky
9 Limits Involving Infinity ∞ (2.5)
Consider following limit: lim
x →0
1
. The graph and table (use
x2
15
convenient values like f (10− n ) = 102 n ) suggests that there’s no
10
number that the function reach as x approach 0. However one
can see that the closer x to the zero (from either side) the
5
0
-5
-4
larger values the function takes. We’ll denote such situation as lim
x →0
-3
-1
-2
0
2
1
3
4
5
1
= ∞ . The
x2
symbol ∞ doesn’t represent a number, but express a particular sort of non-existing
limit.
Def: The line x=a called a vertical asymptote of the curve y=f(x) if at least one of the
following statements is true: lim = ∞,
lim = ∞,
lim = −∞,
lim = −∞
x→a±
Ex 1.
Ex 2.
Ex 3.
x→a
x→a±
x→a
lim ln x = −∞
x →0
lim tan x = ∞
x →π /2−
lim+
x →−1
x −1
x −1
1
cos 2 x = lim+
cos 2 x = lim
cos 2 x = ∞ near x=-1
2
x →−1 x + 1+
x →−1 ( x + 1)( x − 1)
x −1
,
cos 2 x < 1 therefore
cos 2 x
1
≤
→ ∞ . Another reason inf*bounded function.
x +1 x +1
Def: Let f(x) be defined on ( a, ∞ ) , then lim f ( x ) = L means that f(x) gets close to L as
x →±∞
x (or –x) gets sufficiently large. We denote the line y=L as horizontal asymptote to
y=f(x).
2
1.5
1
Ex 4.
y=
e − x ⇒ lim y =
0
Ex 5.
y= arctan x ⇒ lim y= π / 2
Ex 6.
y =1/ x ⇒ lim y =0
Ex 7.
a) Let f ( x) = 
0.5
x →∞
0
x →∞
n
x →±∞
-0.5
-1
-1.5
-2
-20
-15
-10
-5
0
5
10
15
20
x x∈
, lim f ( x ) =
NA, lim f ( n ) =
∞, lim f ( n / π ) =
1
n →∞
n →∞
 1 x ∈  \  x →∞
1 x ∈ 
b) Let g ( x) = 
x
2
x∈ \ 
, again lim g ( x ) = NA , but lim f ( x ) g ( x ) = ∞ , not that
x →∞
x →∞
neither f(x) or g(x) tends to infinite, but their multiplication does.
Ex 8.
lim x sin x = ∞ since sin x ≤ 1 ⇒ x=
sin x x sin x ≤ x → ∞
x →∞
Course: Accelerated Engineering Calculus I
Instructor: Michael Medvinsky
9.1 The arithmetic of Infinite Limits
1. Consider that lim
f ( x) =
x→a
∞
and
lim g ( x ) = ∞
x→a
and, then the following is
true:
lim ( f ( x ) + g ( x ) ) =
∞
x→a
but lim ( f ( x ) − g ( x ) ) =
??? need more work
x→a
lim f ( x ) g ( x ) = ∞
x→a
lim f ( x )
x→a
but lim
x→a
g( x)
= ∞
f ( x)
= ??? need more work
g ( x)
lim ( ± p ) f ( x ) = ±∞,   p > 0
x→a
lim ( f ( x ) ) =
∞,
q > 0
q
x→a
2. Consider that lim
f ( x) =
x→a
∞
and
lim h ( x ) =
± L,   L > 0 ,
∞
and
lim h ( x ) = 0
x→a
true:
lim ( f ( x ) + h ( x ) ) =
∞
x→a
lim f ( x ) h ( x ) = ±∞
x→a
3. Consider that lim
f ( x) =
x→a
lim f ( x ) h ( x ) = ??? need more work
x→a
lim f ( x )
x→a
but lim
x→a
h ( x )
= ??? need more work
f ( x)
= ??? need more work
h ( x )
x→a
then
then the following is