A Deep Dive Into The World Of Limits: Limit Laws

1. Sum Law (Addition Law)

• Law: The limit of the sum of two functions is the sum of their limits.
• Formula: \( \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \).
• Example: \( \lim_{x \to 2} [x^2 + 3x] = \lim_{x \to 2} x^2 + \lim_{x \to 2} 3x = 2^2 + 3(2) = 4 + 6 = 10 \).

2. Difference Law (Subtraction Law)

• Law: The limit of the difference between two functions is the difference of their limits.
• Formula: \( \lim_{x \to a} [f(x) – g(x)] = \lim_{x \to a} f(x) – \lim_{x \to a} g(x) \).
• Example: \( \lim_{x \to 3} [x^3 – x] = \lim_{x \to 3} x^3 – \lim_{x \to 3} x = 3^3 – 3 = 27 – 3 = 24 \).

3. Product Law

• Law: The limit of the product of two functions is the product of their limits.
• Formula: \( \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \).
• Example: \( \lim_{x \to 4} [x \cdot \sin(x)] = \lim_{x \to 4} x \cdot \lim_{x \to 4} \sin(x) = 4 \cdot \sin(4) \).

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4. Quotient Law

• Law: The limit of the quotient of two functions is the quotient of their limits, provided the limit of the denominator is not \(zero\).
• Formula: \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \), where \( \lim_{x \to a} g(x) \neq 0 \).
• Example: \( \lim_{x \to 2} \frac{x^2}{x + 1} = \frac{\lim_{x \to 2} x^2}{\lim_{x \to 2} (x + 1)} = \frac{4}{3} \).

5. Constant Multiple Law

• Law: The limit of a constant multiplied by a function is the constant multiplied by the limit of the function.
• Formula: \( \lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x) \).
• Example: \( \lim_{x \to 5} [3 \cdot \cos(x)] = 3 \cdot \lim_{x \to 5} \cos(x) = 3 \cdot \cos(5) \).

6. Power Law

• Law: The limit of a function raised to a power is the limit of the function raised to that power.
• Formula: \( \lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n \).
• Example: \( \lim_{x \to 2} (x + 1)^3 = [\lim_{x \to 2} (x + 1)]^3 = 3^3 = 27 \).

7. Root Law

• Law: The limit of the nth root of a function is the nth root of the limit of the function.
• Formula: \( \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} \).
• Example: \( \lim_{x \to 4} \sqrt{x + 5} = \sqrt{\lim_{x \to 4} (x + 5)} = \sqrt{9} = 3 \).

8. Constant Law of Limits

• Law: If a function \( f(x) \) is equal to a constant \( b \) for every value of \( x \), then the limit of \( f(x) \) as \( x \) approaches any value \( a \) is also \( b \).
• Formula: If \( f(x) = b \) for all \( x \), then \( \lim_{x \to a} f(x) = b \).
• Example: Let \( f(x) = 5 \) for all \( x \). Then, \( \lim_{x \to 10} f(x) = \lim_{x \to 10} 5 = 5 \).

FAQ

What are the limit laws in calculus?

Limit laws are rules that let you split a complicated limit into simple pieces. If the limits of f(x) and g(x) both exist, the limit of their sum, difference, product or constant multiple comes from the separate limits, and so does a quotient when the bottom limit is not 0. For example, the limit of x² + 3x as x → 2 is 4 + 6 = 10.

How many limit laws are there?

Textbooks group them differently, so you may see anywhere from about five to eleven. The common list covers the sum, difference, product, quotient, constant multiple, power, root and constant laws, sometimes plus the limit of x itself. The exact count matters less than knowing that each law needs the individual limits to exist first.

When can limit laws be applied?

Only when the limits of the pieces exist as finite numbers, and for the quotient law, only when the limit of the denominator is not zero. If substitution gives 0/0, the quotient law cannot be used yet, so simplify first. For (x² − 4)/(x − 2) as x → 2, factor and cancel to get x + 2, then apply the laws to get 4.

Can limits be multiplied?

Yes. By the product law, the limit of f(x)·g(x) equals the limit of f(x) times the limit of g(x), as long as both limits exist. For example, the limit of x(x + 1) as x → 3 is 3 × 4 = 12. If one of the limits does not exist, the product law cannot be used, although the product itself may still have a limit.

Do limit laws work for limits at infinity?

Mostly yes, as long as each piece has a finite limit. As x → ∞, 1/x → 0, so the limit of 5 + 2/x is 5 + 0 = 5. Trouble starts with forms like ∞ − ∞ or ∞/∞, which the laws cannot handle directly. For a rational function, divide every term by the highest power of x first, then apply the laws.

What is the power law for limits?

The power law says the limit of [f(x)]ⁿ equals [lim f(x)]ⁿ for any positive integer n. For example, the limit of (x + 2)³ as x → 1 is 3³ = 27. The root law works the same way, with the limit of the nth root of f(x) equal to the nth root of the limit, as long as that limit is positive when n is even.

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