Limits

Limits

Limits are the soul of calculus, describing the behavior of a function near a point.

Definition and Properties of Limits

• Function Limit: When $x$ approaches $x_0$, $f(x)$ approaches $A$, denoted as $\lim_{x \to x_0} f(x) = A$.
• Sequence Limit: When $n$ approaches infinity, $x_n$ approaches $A$, denoted as $\lim_{n \to \infty} x_n = A$.
• Left and Right Limits: $\lim_{x \to x_0^-} f(x)$ and $\lim_{x \to x_0^+} f(x)$. The limit of $f(x)$ at $x_0$ exists if and only if both the left and right limits exist and are equal.
• Properties of Limits: Uniqueness, boundedness, sign-preserving property.

Infinitesimals and Infinities

• Infinitesimal: If $\lim f(x) = 0$, then $f(x)$ is called an infinitesimal.
• Infinity: If $\lim f(x) = \infty$, then $f(x)$ is called infinite.
• Relationship: In the same process, if $f(x)$ is infinite, then $1/f(x)$ is infinitesimal; conversely, if $f(x)$ is infinitesimal and $f(x) \neq 0$, then $1/f(x)$ is infinite.
• Comparison of Infinitesimals: Let $\lim \alpha(x) = 0$, $\lim \beta(x) = 0$
  • Higher-order infinitesimal: $\lim \frac{\alpha}{\beta} = 0$
  • Lower-order infinitesimal: $\lim \frac{\alpha}{\beta} = \infty$
  • Same-order infinitesimal: $\lim \frac{\alpha}{\beta} = C \neq 0$
  • Equivalent infinitesimal: $\lim \frac{\alpha}{\beta} = 1$, denoted as $\alpha \sim \beta$. Using equivalent infinitesimals is an important method for calculating limits.

Calculation Rules for Limits

If $\lim f(x) = A$, $\lim g(x) = B$, then:

1. $\lim [f(x) \pm g(x)] = A \pm B$
2. $\lim [f(x) \cdot g(x)] = A \cdot B$
3. $\lim \frac{f(x)}{g(x)} = \frac{A}{B}$ (where $B \neq 0$)

Criteria for the Existence of Limits and Two Important Limits

Criteria

1. Squeeze Theorem: If in a neighborhood of a point, $g(x) \leq f(x) \leq h(x)$ always holds, and $\lim g(x) = \lim h(x) = A$, then $\lim f(x) = A$.
2. Monotone Bounded Theorem: A monotone bounded sequence must have a limit.

Two Important Limits

$\lim_{x \to 0} \frac{\sin x}{x} = 1$

$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$

Common Equivalent Infinitesimals

As $x \to 0$:

• $\sin x \sim x$
• $\tan x \sim x$
• $\arcsin x \sim x$
• $\arctan x \sim x$
• $1 - \cos x \sim \frac{x^2}{2}$
• $e^x - 1 \sim x$
• $\ln(1 + x) \sim x$
• $(1 + x)^\alpha - 1 \sim \alpha x$

These equivalent infinitesimals are very useful for calculating limits and can greatly simplify the process.

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Exercises

1. Calculate the limit $\lim\limits_{x \to 0} \frac{\sin 3x}{x}$.
2. Determine whether the sequence $x_n = \frac{1}{n}$ is an infinitesimal sequence.
3. Use equivalent infinitesimals to calculate the limit $\lim\limits_{x \to 0} \frac{1 - \cos x}{x^2}$.
4. Calculate the limit $\lim\limits_{x \to \infty} \left(1 + \frac{2}{x}\right)^x$.
5. Determine the limit of $f(x) = \frac{1}{x}$ as $x \to 0^+$, and state whether it is infinite or infinitesimal.

Reference Answers

1. Calculate $\lim\limits_{x \to 0} \frac{\sin 3x}{x}$

Solution: $\frac{\sin 3x}{x} = 3 \cdot \frac{\sin 3x}{3x}$, as $x \to 0$, $\frac{\sin 3x}{3x} \to 1$.

Answer: The limit is $3$.

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2. Is $x_n = \frac{1}{n}$ an infinitesimal sequence?

Solution: $\lim\limits_{n \to \infty} \frac{1}{n} = 0$.

Answer: Yes, it is an infinitesimal sequence.

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3. Use equivalent infinitesimals to calculate $\lim\limits_{x \to 0} \frac{1 - \cos x}{x^2}$

Solution: $1 - \cos x \sim \frac{x^2}{2}$, so the limit $\approx \frac{\frac{x^2}{2}}{x^2} = \frac{1}{2}$.

Answer: The limit is $\frac{1}{2}$.

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4. Calculate $\lim\limits_{x \to \infty} \left(1 + \frac{2}{x}\right)^x$

Solution: Use the important limit $\lim\limits_{x \to \infty} \left(1 + \frac{a}{x}\right)^x = e^a$. Here $a = 2$, so the limit is $e^2$.

Answer: The limit is $e^2$.

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5. Determine the limit of $f(x) = \frac{1}{x}$ as $x \to 0^+$, and state whether it is infinite or infinitesimal

Solution: As $x \to 0^+$, $f(x) = \frac{1}{x} \to +\infty$, which is infinite.

Answer: The limit does not exist (tends to $+\infty$), it is infinite.

上一章节 Functions Learn the basic concepts, representations, and common properties of functions, such as monotonicity, parity, periodicity, etc. 下一章节 Continuity Understand the definition of function continuity, classification of discontinuities, and properties of continuous functions on closed intervals.