Limit Notation and Terminology

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

1. $\lim_{x \to 5} (x^2) = 25$ → "The limit of x squared as x approaches 5 is 25"

2. $\lim_{x \to 0^+} \frac{1}{x} = \infty$ → "The limit of 1/x as x approaches 0 from the right is infinity"

3. $\lim_{x \to 2} f(x)$ DNE → "The limit of f(x) as x approaches 2 does not exist"