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.