Limits of Logarithmic Functions

The limit of the logarithmic function is an important formula in limit theory and is frequently used in limit computation.

Basic Form

Idea of the Proof

Use the inverse operation of the exponential limit:

1. Let $t = \ln(1 + x)$, then $1 + x = e^t$
2. As $x \to 0$, $t \to 0$
3. $\lim_{x \to 0} \frac{\ln(1 + x)}{x} = \lim_{t \to 0} \frac{t}{e^t - 1} = 1$

Detailed Derivation

Step 1: Change of variable

• Let $t = \ln(1 + x)$
• Then $1 + x = e^t$
• Hence $x = e^t - 1$

Step 2: Limit conversion As $x \to 0$:

• Since $t = \ln(1 + x)$, as $x \to 0$, $1 + x \to 1$, so $t \to 0$

Step 3: Establish the equality $\lim_{x \to 0} \frac{\ln(1 + x)}{x} = \lim_{t \to 0} \frac{t}{e^t - 1}$

Step 4: Use the exponential limit Exponential limit: $\lim_{t \to 0} \frac{e^t - 1}{t} = 1$

The origin of the exponential limit:

This exponential limit is actually derived from the second important limit:

1. The second important limit: $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

2. Through a change of variable: $\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$

3. Taking logarithms on both sides: $\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$

4. Using the inverse relationship between the exponential and logarithmic functions: $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$

Therefore: $\lim_{x \to 0} \frac{\ln(1 + x)}{x} = \lim_{t \to 0} \frac{t}{e^t - 1} = 1$

Key insight: This is the inverse operation of the exponential limit. Through a change of variable, the limit of the logarithmic function is transformed into the limit of the exponential function, and the result is obtained using the known exponential limit.

Generalized Form

$\lim_{x \to 0} \frac{\ln(1 + kx)}{x} = k$

Equivalent Infinitesimal

Using this important limit, we obtain an important equivalent infinitesimal:

• As $x \to 0$, $\ln(1 + x) \sim x$

Worked Examples

Example 1

Find $\lim_{x \to 0} \frac{\ln(1 + 3x)}{x}$

Solution: $\lim_{x \to 0} \frac{\ln(1 + 3x)}{x} = 3$

---

Practice Problems

Exercise 1

Find the limit $\lim_{x \to 0} \frac{\ln(1 + 2x)}{x}$.

Exercise 2

Find the limit $\lim_{x \to 0} \frac{\ln(1 + 4x)}{2x}$.

---

Summary

Symbols Used in This Article

中英对照