Limit Notation and Terminology

Master the formal notation and vocabulary used when working with limits

Limit Notation Explained

Understanding the symbols and language of limits is crucial for reading and writing mathematics correctly.

The Basic Notation

$\lim_{x \to a} f(x) = L$

Breaking this down:

$\lim$ : The limit symbol (abbreviation of "limit")
$x \to a$ : "x approaches a" (shown below the limit symbol)
$f(x)$ : The function we're examining
$= L$ : Equals the limit value L

Reading Limits Aloud

You can say any of these:

"The limit of f(x) as x approaches a equals L"
"The limit of f(x) as x tends to a is L"
"As x approaches a, f(x) approaches L"

One-Sided Notation

Left-hand limit (approaching from the left): $\lim_{x \to a^-} f(x) = L$

The superscript minus sign means "from values less than a"

Right-hand limit (approaching from the right): $\lim_{x \to a^+} f(x) = L$

The superscript plus sign means "from values greater than a"

Infinite Limits

Limit equals infinity: $\lim_{x \to a} f(x) = \infty$ This means f(x) grows without bound as x approaches a.

Limit as x approaches infinity: $\lim_{x \to \infty} f(x) = L$ This means as x gets arbitrarily large, f(x) approaches L.

Important Phrases

| Phrase | Notation | Meaning | |--------|----------|---------| | "x approaches a" | $x \to a$ | x gets closer to a | | "from the left" | $x \to a^-$ | x < a, moving toward a | | "from the right" | $x \to a^+$ | x > a, moving toward a | | "f(x) approaches L" | $f(x) \to L$ | Function values near L | | "does not exist" | DNE | No single value |

Example Statements

$\lim_{x \to 5} (x^2) = 25$ → "The limit of x squared as x approaches 5 is 25"
$\lim_{x \to 0^+} \frac{1}{x} = \infty$ → "The limit of 1/x as x approaches 0 from the right is infinity"
$\lim_{x \to 2} f(x)$ DNE → "The limit of f(x) as x approaches 2 does not exist"

📚 Practice Problems

1Problem 1easy

❓ Question:

Write the following statement using proper limit notation: "As x approaches 4, the function 2x + 3 approaches 11"

💡 Show Solution

The statement translates to:

$\lim_{x \to 4} (2x + 3) = 11$

Breaking it down:

"as x approaches 4" → $x \to 4$
"the function 2x + 3" → $f(x) = 2x + 3$
"approaches 11" → $= 11$

2Problem 2medium

❓ Question:

What is the difference between $\lim_{x \to 3^-} f(x)$ and $\lim_{x \to 3^+} f(x)$ ?

💡 Show Solution

$\lim_{x \to 3^-} f(x)$ is the left-hand limit:

We approach 3 from values less than 3 (like 2.9, 2.99, 2.999...)
We're coming from the left side on a number line

$\lim_{x \to 3^+} f(x)$ is the right-hand limit:

We approach 3 from values greater than 3 (like 3.1, 3.01, 3.001...)
We're coming from the right side on a number line

The overall limit $\lim_{x \to 3} f(x)$ exists only if both one-sided limits exist and are equal.

3Problem 3easy

❓ Question:

Write in limit notation: "The limit of 3x + 1 as x approaches 2 is 7"

💡 Show Solution

Step 1: Identify the components: • Function: f(x) = 3x + 1 • Approaching: x → 2 • Limit value: 7

Step 2: Write in limit notation: lim(x→2) (3x + 1) = 7

Answer: lim(x→2) (3x + 1) = 7

4Problem 4medium

❓ Question:

What is the difference between lim(x→3⁻) f(x) and lim(x→3⁺) f(x)?

💡 Show Solution

Step 1: Explain left-hand limit (x→3⁻): The superscript "−" means from the LEFT We approach 3 from values less than 3 (like 2.9, 2.99, 2.999...)

Step 2: Explain right-hand limit (x→3⁺): The superscript "+" means from the RIGHT We approach 3 from values greater than 3 (like 3.1, 3.01, 3.001...)

Step 3: When do they matter? • If lim(x→3⁻) f(x) = lim(x→3⁺) f(x), then lim(x→3) f(x) exists • If they're different, the two-sided limit DNE

Step 4: Example: For f(x) = |x|/x at x = 0: • lim(x→0⁻) f(x) = -1 (from left) • lim(x→0⁺) f(x) = 1 (from right) • lim(x→0) f(x) DNE (they don't match)

Answer: x→3⁻ approaches from the left; x→3⁺ approaches from the right

5Problem 5hard

❓ Question:

Interpret: lim(x→∞) (1/x) = 0

💡 Show Solution

Step 1: Break down the notation: • x→∞ means x gets arbitrarily large (positive) • 1/x is the function • = 0 is the limiting value

Step 2: Numerical analysis: x: 10 100 1000 10000 ... 1/x: 0.1 0.01 0.001 0.0001 ...

Step 3: Interpretation: As x becomes larger and larger (approaching infinity), 1/x becomes smaller and smaller (approaching 0)

Step 4: Graphical meaning: The graph of y = 1/x has a horizontal asymptote at y = 0

Step 5: Important note: x never actually "reaches" infinity We're describing the trend as x grows without bound

Answer: As x increases without bound, 1/x approaches 0

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