We study the base e of natural logarithms and important related limits.
Behavior of Exponential and Logarithmic Functions
- a>1：lim_{x→∞} a^x = ∞、lim_{x→−∞} a^x = 0
- 0
- a>1：lim_{x→∞} log_a x = ∞、lim_{x→+0} log_a x = −∞
Base e of Natural Logarithms
e = lim_{n→∞} (1+1/n)^n = lim_{h→0} (1+h)^{1/h}（≒2.71828）
- lim_{x→0} (e^x − 1)/x = 1
- lim_{x→0} log(1+x)/x = 1（natural logarithm）
- lim_{x→∞} (1+a/x)^x = e^a
Example ①
(1) lim_{x→0} (e^{2x}−1)/x (2) lim_{x→0} log(1+3x)/x
Solution:(1) 2·(e^{2x}−1)/(2x) → 2 (2) 3
(1) lim_{x→0} (e^{2x}−1)/x (2) lim_{x→0} log(1+3x)/x
Solution:(1) 2·(e^{2x}−1)/(2x) → 2 (2) 3
Indeterminate Form 1^∞
Transform the expression into the form (1+a/x)^x.
Example：(1+2/x)^x = {(1+2/x)^{x/2}}² → e²
Example ②
lim_{x→∞} (1+2/x)^x
Solution:u=x/2：{(1+1/u)^u}² → e²
lim_{x→∞} (1+2/x)^x
Solution:u=x/2：{(1+1/u)^u}² → e²
Key Points
- e = lim(1+1/n)^n
- (e^x−1)/x → 1、log(1+x)/x → 1
- (1+a/x)^x → e^a
- Substitute the “inside” expression with u to match the formula
Practice Problems
- lim_{x→0} (e^{5x}−1)/(2x)
- lim_{x→∞} (1+3/x)^x
- lim_{x→0} log(1+2x)/sin x
Answers and Explanations
- Answer:5/2
(5/2)·(e^{5x}−1)/(5x) → 5/2。 - Answer:e³
{(1+3/x)^{x/3}}³ → e³。 - Answer:2
{log(1+2x)/(2x)·2}/{sin x/x} → 2/1 = 2。