Applications of Important Limits

Important limits are not only the foundation of mathematical theory, but also have wide applications in practice. Mastering these applications can help us better understand the essence of limits.

1. Equivalent Infinitesimals

Using important limits, we can obtain important equivalent infinitesimals, which are very useful in limit computation.

Basic Equivalent Infinitesimals

Applications of Equivalent Infinitesimals

Example 1: Find $\lim_{x \to 0} \frac{e^x - 1}{\sin x}$

Solution:

• As $x \to 0$, $e^x - 1 \sim x$ and $\sin x \sim x$
• Therefore $\lim_{x \to 0} \frac{e^x - 1}{\sin x} = \lim_{x \to 0} \frac{x}{x} = 1$

Example 2: Find $\lim_{x \to 0} \frac{\ln(1 + x^2)}{x^2}$

Solution:

• As $x \to 0$, $\ln(1 + x^2) \sim x^2$
• Therefore $\lim_{x \to 0} \frac{\ln(1 + x^2)}{x^2} = 1$

2. Limit Computation

Important limits are fundamental tools for computing complicated limits.

Limits of Composite Functions

Example 1: Find $\lim_{x \to 0} \frac{\sin(e^x - 1)}{x}$

Solution:

1. As $x \to 0$, $e^x - 1 \sim x$
2. Hence $\sin(e^x - 1) \sim e^x - 1 \sim x$
3. $\lim_{x \to 0} \frac{\sin(e^x - 1)}{x} = \lim_{x \to 0} \frac{x}{x} = 1$

Example 2: Find $\lim_{x \to 0} \frac{e^{\sin x} - 1}{x}$

Solution:

1. As $x \to 0$, $\sin x \sim x$
2. Hence $e^{\sin x} - 1 \sim \sin x \sim x$
3. $\lim_{x \to 0} \frac{e^{\sin x} - 1}{x} = \lim_{x \to 0} \frac{x}{x} = 1$

Limits of Power-Exponential Functions

Example 1: Find $\lim_{x \to 0} (1 + x)^{\frac{1}{x}}$

Solution:

• This is the reciprocal form of the second important limit
• $\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$

Example 2: Find $\lim_{x \to \infty} \left(1 + \frac{1}{x^2}\right)^{x^2}$

Solution:

• Let $t = x^2$; then as $x \to \infty$, $t \to \infty$
• $\lim_{x \to \infty} \left(1 + \frac{1}{x^2}\right)^{x^2} = \lim_{t \to \infty} \left(1 + \frac{1}{t}\right)^t = e$

3. Financial Applications

The second important limit has important applications in finance.

Continuously Compounded Interest

In finance, the formula for continuously compounded interest is: $A = P \cdot e^{rt}$

where:

• $A$ is the final amount
• $P$ is the principal
• $r$ is the annual interest rate
• $t$ is the time (in years)

Example: If the principal is 1000 yuan, the annual interest rate is 5%, and interest is compounded continuously for 3 years, find the final amount.

Solution: $A = 1000 \cdot e^{0.05 \times 3} = 1000 \cdot e^{0.15} \approx 1161.83$

Present Value

The present value formula is: $P = A \cdot e^{-rt}$

Example: If 1000 yuan is needed in 3 years and the annual interest rate is 5%, find the present value.

Solution: $P = 1000 \cdot e^{-0.05 \times 3} = 1000 \cdot e^{-0.15} \approx 860.71$

4. Physical Applications

Important limits also have applications in physics.

Instantaneous Velocity

The instantaneous velocity is defined as: $v = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t}$

This is essentially a limiting process.

Instantaneous Acceleration

The instantaneous acceleration is defined as: $a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t}$

5. Engineering Applications

Signal Processing

In signal processing, the limiting properties of the sine function are used to analyze the frequency characteristics of signals.

Control Systems

In control systems, limits are used to analyze the stability and response characteristics of systems.

Practice Problems

Exercise 1

Find the limit $\lim_{x \to 0} \frac{e^x - 1}{\sin x}$.

Exercise 2

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

Exercise 3

Find the limit $\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^x$.

Exercise 4

If the principal is 2000 yuan, the annual interest rate is 6%, and interest is compounded continuously for 5 years, find the final amount.

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Summary

Symbols Used in This Article

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