1/26/2013
Math 103 – Rimmer
2.6 Limits at Infinity;
Horizontal Asymptotes
lim f ( x ) = L
some finite
number
⇒ y = L is a horizontal
x →∞
asymptote of f ( x )
lim f ( x ) = M
⇒ y = M is a horizontal
x → −∞
some finite
asymptote of f ( x )
number
lim arctan x =
x →∞
y = arctan x
lim arctan x =
x →−∞
g ( x)
=2
= −2
=∞
= −∞
= −∞
π
2
−π
2
Math 103 – Rimmer
2.6 Limits at Infinity;
Horizontal Asymptotes
y=2
y = −2
x=3
x=0
x = −2
1
1/26/2013
Math 103 – Rimmer
2.6 Limits at Infinity;
Horizontal Asymptotes
From section 2.3:
Let k and a be constants. Let lim f ( x ) = L and lim g ( x ) = M
x→a
x→a
( with L and M
finite and M ≠ 0 )
1) lim  f ( x ) + g ( x )  = lim f ( x ) + lim g ( x ) = L + M
x→a
x →a
x→a
2 ) lim  f ( x ) − g ( x )  = lim f ( x ) − lim g ( x ) = L − M
x→a
x →a
x →a
3) lim  f ( x ) ⋅ g ( x )  = lim f ( x ) ⋅ lim g ( x ) = LM
x →a
x →a
x→ a
4 ) lim
x→a
f ( x)
g ( x)
=
lim f ( x )
x →a
lim g ( x )
=
L
M
( the key here is that M ≠ 0 )
n
x→a
5 ) lim kf ( x ) = k ⋅ lim f ( x ) = kL
x →a
All these hold when you
replace x → a with x → ∞
n
8 ) lim  f ( x )  = lim f ( x )  = Ln
x→a
 x→a

x→a
n any positive integer
6 ) lim k = k
x→a
7 ) lim x = a
x→a
9 ) lim n f ( x ) = n lim f ( x )
x→a
x→a
n any positive integer
If n is even, assume lim f ( x ) > 0
x→a
Math 103 – Rimmer
2.6 Limits at Infinity;
Horizontal Asymptotes
1
=0
x →∞ x
1
lim = 0
x → −∞ x
lim
If r > 0 is a rational number, then lim
x →∞
1
=0
xr
If r > 0 is a rational number such that x r is defined for all x,
1
then lim r = 0
x → −∞ x
2
1/26/2013
Math 103 – Rimmer
2.6 Limits at Infinity;
Horizontal Asymptotes
3x2 − x + 4
x →∞ 2 x 2 + 5 x − 8
lim
Algebraic way of solving this problem:
1. Find the highest power of x
in the denominator, call it p
1
2. Multiply numerator and
x2
= lim
1
x →∞
1
2
denominator by p
2
x
+
5
x
−
8
(
) x2
x
0
0
1 4
1
4
3− + 2
lim 3 − lim + lim 2
3
x →∞ x
x →∞ x
x x = x →∞
= lim
=
x →∞
5 8
5 0
8 0
2
2+ − 2
lim 2 + lim − lim 2
x →∞
x →∞ x
x →∞ x
x x
( 3x 2 − x + 4 )
If f ( x ) is a rational function,
with deg. num. = deg. denom.
⇒ lim f ( x ) =
x →±∞
coeff. of leading term in num.
coeff. of leading term in denom.
Math 103 – Rimmer
2.6 Limits at Infinity;
Horizontal Asymptotes
x2 + 4
x → −∞ x 2 + x 3 − 1
lim
(x
= lim
x → −∞
(x
2
2
+ 4)
1
x3
+ x 3 − 1)
1
x3
0
0
1 4
1
4
+ 3
lim + lim 3
x → −∞ x
x → −∞ x
= lim x x
= lim
0 = 0
0
x → −∞ 1
x
→
−∞
1
1
1
+1− 3
lim + lim 1 − lim 3
x → −∞ x
x → −∞
x → −∞ x
x
x
If f ( x ) is a rational function,
with deg. num. < deg. denom.
⇒ lim f ( x ) = 0
x →±∞
3
1/26/2013
Math 103 – Rimmer
2.6 Limits at Infinity;
Horizontal Asymptotes
x3 − 2 x + 3
x →∞
5 − 2 x2
lim
0
2 3
x− + 2
( x − 2 x + 3) x12
x x = lim x = −∞
= lim
= lim
x →∞ −2
x →∞
x →∞
50
−2
( 5 − 2 x2 ) x12
2
x
3
If f ( x ) is a rational function,
with deg. num. > deg. denom.
then lim f ( x ) could be ∞ or − ∞ based on
x →±∞
a) the leading coefficients,
b) the leftover power of x in the
numerator being odd or even,
c) whether the limit is going to ∞ or − ∞
Math 103 – Rimmer
2.6 Limits at Infinity;
Horizontal Asymptotes
lim
x → −∞
= lim
x → −∞
9 x6 − x
x3 + 1
(
x3 leading term
in denominator
since x → −∞, x 3 is negative
⇒ x 6 must be negative
9 x6 − x
1
9 x6 − x
9x − x 3
x = lim
x3
− x6
=
lim
( x3 + 1) x13 x→ −∞ 1 + x13 x→ −∞ 1 + 13
x
6
)
10
9 x6 − x
−
9
−
−
x5 = − 9
x6
= lim
= lim
x → −∞
10
x → −∞
1
1
1+ 3
1+ 3
x
x
= −3
4
1/26/2013
Math 103 – Rimmer
2.6 Limits at Infinity;
Horizontal Asymptotes
lim 9 x 2 + x − 3 x
x→ ∞
(
= lim
9 x 2 + x − 3x
1
x→ ∞
= lim
x→ ∞
x
2
9 x + x + 3x
)(
(
) = lim (9 x + x ) − 9 x
9 x + x + 3x
9 x + x + 3x )
9 x 2 + x + 3x
x⋅
x→ ∞
(
1
9 x + x + 3x
x
= lim
x→ ∞
)
2
9x2 + x ∼ x
x is the leading term
in the denominator
1
x
1
2
9x + x
+3
x2
2
2
x→ ∞
2
= lim
2
= lim
= lim
x→ ∞
x→ ∞
1
= lim
2
9x + x
+3
x
x→ ∞
1
2
9x + x
x2
+3
1
1
=
0
9 +3
1
9+ +3
x
=
1
6
5