Calculus- Early Transcendentals, 2021a

3.5. Infinite Limits and Limits at Infinity 89
Example 3.21: Limit at Infinity and Basic Functions
We can easily evaluate the following limits by observation:
6
1. lim √ = 0 2. lim x −
x→∞ x 3 x→−∞ x2 = −∞
3. lim
x→∞
x 3 + x = ∞
4. lim
x→∞
cos(x)=DNE
5. lim
x→∞
e x = ∞ 6. lim
x→−∞ ex = 0
7. lim lnx = −∞
x→0 +
8. lim cos(1/x)=DNE
x→0
Often, the shorthand notation 1
1
=+∞ and = −∞ is used to represent the following two limits
0 + 0− respectively:
1
1
lim =+∞ and lim
x→0 + x x→0 − x = −∞.
Using the above convention we can compute the following limits.
Example 3.22: Limit at Infinity and Basic Functions
Compute lim
x→0 + e1/x , lim
x→0 − e1/x and lim
x→0
e 1/x .
Solution. We have:
lim e 1 1
x = e 0 + = e +∞ = ∞.
x→0 +
lim e 1 1
x = e 0 − = e −∞ = 0.
x→0 −
Thus, as left-hand limit ≠ right-hand limit,
3.5.1 Vertical Asymptotes
lim e 1 x = DNE.
x→0
The line x = a is called a vertical asymptote of f (x) if at least one of the following is true:
♣
lim
x→a f (x)=∞
lim
x→a f (x)=−∞
lim f (x)=∞
x→a −
lim f (x)=−∞
x→a −
lim f (x)=∞
x→a +
lim f (x)=−∞
x→a +