Limits of Power Functions

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

Basic Form

Idea of the Proof

Use the properties of the logarithmic and exponential functions:

1. $(1 + x)^\alpha = e^{\alpha \ln(1 + x)}$
2. As $x \to 0$, $\ln(1 + x) \sim x$
3. Therefore $(1 + x)^\alpha - 1 \sim \alpha x$

Detailed Derivation

Step 1: Use the relation between exponential and logarithm $(1 + x)^\alpha = e^{\alpha \ln(1 + x)}$

Step 2: Use the equivalent infinitesimal As $x \to 0$, $\ln(1 + x) \sim x$ (logarithmic limit)

Step 3: Derive the power limit $\lim_{x \to 0} \frac{(1 + x)^\alpha - 1}{x} = \lim_{x \to 0} \frac{e^{\alpha \ln(1 + x)} - 1}{x}$

As $x \to 0$, $\alpha \ln(1 + x) \sim \alpha x$, so: $\lim_{x \to 0} \frac{e^{\alpha x} - 1}{x} = \alpha$

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

The origin of the exponential limit:

This exponential limit is 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: $\lim_{x \to 0} \frac{(1 + x)^\alpha - 1}{x} = \alpha$

Key insight: The power limit can be derived through the properties of the exponential and logarithmic functions. When the exponent $\alpha$ is an integer, it can also be proved by expanding with the binomial theorem.

Special Case

When $\alpha = 1$: $\lim_{x \to 0} \frac{(1 + x) - 1}{x} = 1$

Equivalent Infinitesimal

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

• As $x \to 0$, $(1 + x)^\alpha - 1 \sim \alpha x$

Worked Examples

Example 1

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

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

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Practice Problems

Exercise 1

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

Exercise 2

Find the limit $\lim_{x \to 0} \frac{(1 + 2x)^5 - 1}{x}$.

Exercise 3

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

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Summary

Symbols Used in This Article

中英对照