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Basic Concepts of Limits

Limits are the soul of calculus. They describe how a function behaves near a point and form the bedrock for learning calculus.

Definition of Limits

Function Limits

When $x$ approaches $x_0$ , $f(x)$ approaches $A$ , denoted as:

\lim_{x \to x_0} f(x) = A

Mathematical wording: For any $\varepsilon > 0$ , there exists a $\delta > 0$ such that whenever $0 < \lvert x - x_0 \rvert < \delta$ , we have $\lvert f(x) - A \rvert < \varepsilon$ .

Symbol	Type	Pronunciation/Description	Meaning in this context
$\varepsilon$	Greek letter	Epsilon	Arbitrarily small positive number
$\delta$	Greek letter	Delta	Positive number depending on $\varepsilon$

Sequence Limits

When $n$ approaches infinity, $x_n$ approaches $A$ , denoted as:

\lim_{n \to \infty} x_n = A

Mathematical wording: For any $\varepsilon > 0$ , there exists an integer $N$ such that when $n > N$ , we have $\lvert x_n - A \rvert < \varepsilon$ .

Examples：

$\lim_{n \to \infty} \frac{1}{n} = 0$
$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$

Left and Right Limits

Left Limits

When $x$ approaches $x_0$ from the left, the limit of $f(x)$ is written as:

\lim_{x \to x_0^-} f(x)

Mathematical wording: For any $\varepsilon > 0$ , there exists a $\delta > 0$ such that whenever $x_0 - \delta < x < x_0$ , we have $\lvert f(x) - A \rvert < \varepsilon$ .

Right Limits

When $x$ approaches $x_0$ from the right, the limit of $f(x)$ is written as:

\lim_{x \to x_0^+} f(x)

Mathematical wording: For any $\varepsilon > 0$ , there exists a $\delta > 0$ such that whenever $x_0 < x < x_0 + \delta$ , we have $\lvert f(x) - A \rvert < \varepsilon$ .

Necessary and Sufficient Condition for Limit Existence

The limit of $f(x)$ exists at $x_0$ if and only if its left-hand and right-hand limits both exist and are equal.

$\lim_{x \to x_0} f(x) = A \Leftrightarrow \lim_{x \to x_0^-} f(x) = \lim_{x \to x_0^+} f(x) = A$

Examples：

For $f(x) = \frac{|x|}{x}$ at $x = 0$ , the left-hand limit is $-1$ and the right-hand limit is $1$ , so the limit does not exist.

Properties of Limits

Uniqueness

Uniqueness of Limits

If the limit exists, it is unique.

Proof

Assume $\lim_{x \to x_0} f(x) = A$ and $\lim_{x \to x_0} f(x) = B$ ; then $A = B$ .

Boundedness

Boundedness of Limits

If $\lim_{x \to x_0} f(x) = A$ , then there is a neighborhood of $x_0$ on which $f(x)$ is bounded.

Corollary: If a function has a limit at a point, it must be bounded in some neighborhood of that point.

Sign Preservation

Sign Preservation of Limits

If $\lim_{x \to x_0} f(x) = A > 0$ , then there is a neighborhood of $x_0$ where $f(x) > 0$ .

Corollary: If $\lim_{x \to x_0} f(x) = A < 0$ , then there is a neighborhood of $x_0$ where $f(x) < 0$ .

Geometric Interpretation of Limits

Geometric Meaning of Function Limits

As $x$ gets arbitrarily close to $x_0$ , $f(x)$ gets arbitrarily close to the constant $A$ .
The graph near $x_0$ clusters around the line $y = A$ .
From either side of $x_0$ , $f(x)$ tends to the same value.

Geometric Meaning of Sequence Limits

Points of the sequence on the number line get arbitrarily close to $A$ .
Beyond some term, all points fall inside any neighborhood of $A$ .
The “tail” of the sequence gets closer and closer to its limit.

Exercises

Exercise 1

Determine whether the limit of $f(x) = \frac{x^2 - 1}{x - 1}$ exists at $x = 1$ .

Reference Answer

function limitleft-hand limitright-hand limit

Approach: Compute left-hand and right-hand limits and see if they are equal.

Steps:

Right-hand limit: $\lim_{x \to 1^+} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1^+} \frac{(x-1)(x+1)}{x-1} = \lim_{x \to 1^+} (x+1) = 2$
Left-hand limit: $\lim_{x \to 1^-} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1^-} \frac{(x-1)(x+1)}{x-1} = \lim_{x \to 1^-} (x+1) = 2$
Since the left-hand and right-hand limits match, the limit exists.

Answer:The limit exists and equals $2$ .

Exercise 2

Prove that the sequence $x_n = \frac{n}{n+1}$ has limit $1$ .

Reference Answer

sequence limit

Approach: Use the definition of a limit: for any $\varepsilon > 0$ , find $N$ such that $|x_n - 1| < \varepsilon$ for $n > N$ .

Steps:

$|x_n - 1| = \left|\frac{n}{n+1} - 1\right| = \left|\frac{n-(n+1)}{n+1}\right| = \frac{1}{n+1}$
To make $\frac{1}{n+1} < \varepsilon$ , we need $n+1 > \frac{1}{\varepsilon}$ , i.e., $n > \frac{1}{\varepsilon} - 1$
Let $N = \left\lfloor \frac{1}{\varepsilon} - 1 \right\rfloor + 1$ ; then for $n > N$ , $|x_n - 1| < \varepsilon$

Answer:The sequence limit is $1$ .

Exercise 3

Determine whether the limit of $f(x) = \frac{1}{x}$ exists at $x = 0$ .

Reference Answer

function limitleft-hand limitright-hand limit

Approach: Compute left-hand and right-hand limits.

Steps:

Right-hand limit: $\lim_{x \to 0^+} \frac{1}{x} = +\infty$
Left-hand limit: $\lim_{x \to 0^-} \frac{1}{x} = -\infty$
Since they differ, the limit does not exist.

Answer:The limit does not exist.

Summary

Symbols Used in This Article

Symbol notation

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\varepsilon$	Greek letter	Epsilon (/ˈɛpsələn/)	Arbitrarily small positive number
$\delta$	Greek letter	Delta (/ˈdɛltə/)	Positive number depending on $\varepsilon$
$N$	Mathematical symbol	Positive integer	A sufficiently large positive integer
$\lim$	Mathematical symbol	Limit	Denotes the limit of a function or sequence
$\to$	Mathematical symbol	Tends to	Indicates approaching a value
$\infty$	Mathematical symbol	Infinity	Represents infinity
$\vert x - x_0 \vert$	Mathematical symbol	Absolute value	Distance between $x$ and $x_0$

Operation Rules of Limits

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Basic Concepts of Limits

Definition of Limits

Function Limits

Sequence Limits

Left and Right Limits

Left Limits

Right Limits

Necessary and Sufficient Condition for Limit Existence

Properties of Limits

Uniqueness

Boundedness

Sign Preservation

Geometric Interpretation of Limits

Geometric Meaning of Function Limits

Geometric Meaning of Sequence Limits

Exercises

Exercise 1

Exercise 2

Exercise 3

Summary

Symbols Used in This Article

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