Limit Properties for MATH 122

Exam Relevance for MATH 122

Likelihood of appearing: Essential

Needed to evaluate limits. Rarely tested directly but essential for limit computations.

Lesson

What are Limit Properties?

Limit properties (also called limit laws) are rules that let you break down complicated limits into simpler pieces.

The Idea: Instead of evaluating one big scary limit, you can:

Split it into smaller limits
Evaluate each small limit
Combine the results

These rules work as long as the individual limits exist.

The Basic Limit Properties

Assume $\displaystyle\lim_{x \to a} f(x) = L$ and $\displaystyle\lim_{x \to a} g(x) = M$ both exist. Then:

1. Constant Multiple Rule

$$\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x) = c \cdot L$$

You can pull constants out of limits.

Example: When It Works ✓

Problem: If $\lim_{x \to 3} f(x) = 4$, find $\lim_{x \to 3} 5f(x)$.

$$\lim_{x \to 3} 5f(x) = 5 \cdot \lim_{x \to 3} f(x) = 5 \cdot 4 = \boxed{20}$$

Example: When It Fails ✗

Problem: Find $\lim_{x \to 0} \frac{3}{x^2}$.

Why it fails: You can pull out the 3, giving $3 \cdot \lim_{x \to 0} \frac{1}{x^2}$, but $\lim_{x \to 0} \frac{1}{x^2} = \infty$ (doesn't exist as a finite number). The rule requires the inner limit to exist!

2. Sum/Difference Rule

$$\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) = L \pm M$$

The limit of a sum equals the sum of the limits.

Example: When It Works ✓

Problem: Find $\lim_{x \to 2} (x + x^2)$.

$$\lim_{x \to 2} (x + x^2) = \lim_{x \to 2} x + \lim_{x \to 2} x^2 = 2 + 4 = \boxed{6}$$

Example: When It Fails ✗

Problem: Find $\lim_{x \to \infty} (x - x)$.

Why it fails: You might think $\lim_{x \to \infty} x - \lim_{x \to \infty} x = \infty - \infty$, but $\infty - \infty$ is indeterminate! The rule only works when both limits exist as finite numbers. (The actual answer is 0, but you need to simplify first: $x - x = 0$.)

3. Product Rule

$$\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) = L \cdot M$$

The limit of a product equals the product of the limits.

Example: When It Works ✓

Problem: Find $\lim_{x \to 3} (x \cdot x^2)$.

$$\lim_{x \to 3} (x \cdot x^2) = \lim_{x \to 3} x \cdot \lim_{x \to 3} x^2 = 3 \cdot 9 = \boxed{27}$$

Example: When It Fails ✗

Problem: Find $\lim_{x \to 0} \left(x \cdot \frac{1}{x}\right)$.

Why it fails: You might try $\lim_{x \to 0} x \cdot \lim_{x \to 0} \frac{1}{x} = 0 \cdot \infty$, but $0 \cdot \infty$ is indeterminate! Simplify first: $x \cdot \frac{1}{x} = 1$, so the limit is 1.

4. Quotient Rule

$$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\displaystyle\lim_{x \to a} f(x)}{\displaystyle\lim_{x \to a} g(x)} = \frac{L}{M} \quad \text{(if } M \neq 0\text{)}$$

The limit of a quotient equals the quotient of the limits (as long as the denominator isn't zero).

Example: When It Works ✓

Problem: Find $\lim_{x \to 4} \frac{x^2}{x}$.

$$\lim_{x \to 4} \frac{x^2}{x} = \frac{\lim_{x \to 4} x^2}{\lim_{x \to 4} x} = \frac{16}{4} = \boxed{4}$$

Example: When It Fails ✗

Problem: Find $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$.

Why it fails: Direct substitution gives $\frac{0}{0}$, which is indeterminate. You can't use the quotient rule when the denominator's limit is 0! Factor first: $\frac{(x-2)(x+2)}{x-2} = x + 2$, so the limit is 4.

5. Power Rule

$$\lim_{x \to a} [f(x)]^n = \left[\lim_{x \to a} f(x)\right]^n = L^n$$

The limit of a power equals the power of the limit.

Example: When It Works ✓

Problem: Find $\lim_{x \to 2} (x + 1)^3$.

$$\lim_{x \to 2} (x + 1)^3 = \left(\lim_{x \to 2} (x + 1)\right)^3 = 3^3 = \boxed{27}$$

Example: When It Fails ✗

Problem: Find $\lim_{x \to 0^+} \left(\frac{1}{x}\right)^2$.

Why it fails: You'd need $\left(\lim_{x \to 0^+} \frac{1}{x}\right)^2 = (\infty)^2$, but $\infty$ is not a number you can raise to a power using this rule. The limit is $\infty$, but you determine this by analyzing the function's behavior, not by applying the power rule.

6. Root Rule

$$\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} = \sqrt[n]{L}$$

Works when $n$ is odd, or when $n$ is even and $L \geq 0$.

Example: When It Works ✓

Problem: Find $\lim_{x \to 5} \sqrt{x + 4}$.

$$\lim_{x \to 5} \sqrt{x + 4} = \sqrt{\lim_{x \to 5} (x + 4)} = \sqrt{9} = \boxed{3}$$

Example: When It Fails ✗

Problem: Find $\lim_{x \to 3} \sqrt{x - 5}$.

Why it fails: $\lim_{x \to 3} (x - 5) = -2$, and $\sqrt{-2}$ is not a real number! For even roots, you need $L \geq 0$. This limit does not exist in the real numbers.

Special Simple Limits

These are building blocks you'll use constantly:

$$\lim_{x \to a} c = c \quad \text{(constant)}$$

$$\lim_{x \to a} x = a \quad \text{(identity)}$$

$$\lim_{x \to a} x^n = a^n \quad \text{(power)}$$

Direct Substitution

For polynomial and rational functions (where the denominator isn't zero), you can just plug in the value:

$$\lim_{x \to a} p(x) = p(a)$$

This works because polynomials are continuous everywhere!

Example Problems

Example 1: Using Multiple Properties

Problem: Evaluate $\displaystyle\lim_{x \to 2} (3x^2 - 4x + 5)$

Step 1: Use the Sum/Difference Rule to split it up

$$\lim_{x \to 2} (3x^2 - 4x + 5) = \lim_{x \to 2} 3x^2 - \lim_{x \to 2} 4x + \lim_{x \to 2} 5$$

Step 2: Use the Constant Multiple Rule

$$= 3\lim_{x \to 2} x^2 - 4\lim_{x \to 2} x + 5$$

Step 3: Use the basic limits ($\lim x^n = a^n$)

$$= 3(2)^2 - 4(2) + 5$$ $$= 3(4) - 8 + 5$$ $$= 12 - 8 + 5$$ $$\boxed{= 9}$$

Shortcut: Since this is a polynomial, just plug in $x = 2$: $$3(2)^2 - 4(2) + 5 = 12 - 8 + 5 = 9 \checkmark$$

Example 2: Quotient with Direct Substitution

Problem: Evaluate $\displaystyle\lim_{x \to 3} \frac{x^2 + 1}{x - 1}$

Step 1: Check if the denominator equals zero at $x = 3$

$$3 - 1 = 2 \neq 0 \quad \checkmark$$

Step 2: Since the denominator isn't zero, use the Quotient Rule (or just substitute)

$$\lim_{x \to 3} \frac{x^2 + 1}{x - 1} = \frac{3^2 + 1}{3 - 1} = \frac{10}{2} = \boxed{5}$$

Example 3: Using the Root Rule

Problem: Evaluate $\displaystyle\lim_{x \to 4} \sqrt{x^2 + 9}$

Step 1: Apply the Root Rule

$$\lim_{x \to 4} \sqrt{x^2 + 9} = \sqrt{\lim_{x \to 4} (x^2 + 9)}$$

Step 2: Evaluate the inner limit

$$= \sqrt{4^2 + 9} = \sqrt{16 + 9} = \sqrt{25} = \boxed{5}$$

When Direct Substitution FAILS

Direct substitution fails when you get:

$\frac{0}{0}$ — indeterminate form (need algebraic tricks)
$\frac{\text{number}}{0}$ — usually means infinity or DNE

When this happens, you need other techniques:

Factoring and canceling
Rationalizing (multiply by conjugate)
L'Hôpital's Rule (later topic)

Example 4: When You Can't Just Plug In

Problem: Evaluate $\displaystyle\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$

Step 1: Try direct substitution

$$\frac{2^2 - 4}{2 - 2} = \frac{0}{0} \quad \text{Indeterminate!}$$

Step 2: Factor and simplify

$$\frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2} = x + 2 \quad (\text{for } x \neq 2)$$

Step 3: Now evaluate

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

Formulas & Reference

Constant Multiple Rule

\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)

You can pull constants out of limits. Only works if the limit of f(x) exists as a finite number.

Variables:

$c$:: any constant
$\lim_{x \to a} f(x)$:: must exist (finite)

Sum/Difference Rule

\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)

The limit of a sum (or difference) equals the sum (or difference) of the limits. Only works if both individual limits exist as finite numbers.

Variables:

$\lim_{x \to a} f(x)$:: must exist (finite)
$\lim_{x \to a} g(x)$:: must exist (finite)

Product Rule

\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)

The limit of a product equals the product of the limits. Only works if both individual limits exist as finite numbers.

Variables:

$\lim_{x \to a} f(x)$:: must exist (finite)
$\lim_{x \to a} g(x)$:: must exist (finite)

Quotient Rule

\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}

The limit of a quotient equals the quotient of the limits. Only works if both individual limits exist as finite numbers AND the denominator's limit is not zero.

Variables:

$\lim_{x \to a} f(x)$:: must exist (finite)
$\lim_{x \to a} g(x)$:: must exist (finite) and ≠ 0

Power Rule

\lim_{x \to a} [f(x)]^n = \left[\lim_{x \to a} f(x)\right]^n

The limit of a power equals the power of the limit. Only works if the limit of f(x) exists as a finite number.

Variables:

$\lim_{x \to a} f(x)$:: must exist (finite)
$n$:: any positive integer

Root Rule

\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}

The limit of a root equals the root of the limit. Only works if the limit of f(x) exists as a finite number. For even roots, the limit must also be non-negative.