Topic 1.5 Notes – Determining Limits Using Algebraic Properties of Limits

The Algebraic Properties of Limits

Suppose

$$\lim_{x \to c} f(x) = L \quad \text{and} \quad \lim_{x \to c} g(x) = M.$$

When those limits exist, you can treat them like numbers.

Limit Theorems

• Sum Rule

  $$\lim_{x \to c} (f + g) = L + M$$

• Difference Rule

  $$\lim_{x \to c} (f - g) = L - M$$

• Constant Multiple Rule

  $$\lim_{x \to c} (k f) = kL$$

• Product Rule

  $$\lim_{x \to c} (f \cdot g) = LM$$

• Quotient Rule (as long as $M \ne 0$)

  $$\lim_{x \to c} \frac{f}{g} = \frac{L}{M}$$

• Power Rule (positive integers $n$)

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

• Root Rule

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

The big idea: limits distribute across algebraic operations.

If each piece has a limit, the whole expression usually does too.

Direct Substitution and When It Works

Most of the time in this topic, you’ll just plug in $x = c$.

Direct substitution works immediately for:

• Polynomials
• Rational functions where the denominator is not zero at $x=c$
• Expressions built from sums, products, powers, and roots of “nice” functions

Example:

$$\lim_{x \to 2} (4x^3 - 5x + 1)$$

Just plug in:

$$4(2)^3 - 5(2) + 1 = 32 - 10 + 1 = 23$$

No tricks. Polynomials are continuous everywhere, so substitution works.

Rational Example

$$\lim_{x \to 1} \frac{x^2 + 3}{x + 4}$$

Plug in:

$$\frac{1^2 + 3}{1 + 4} = \frac{4}{5}$$

The denominator isn’t zero, so the quotient rule applies cleanly.

If plugging in gives you a denominator of 0, you cannot use the quotient rule directly. That’s when later techniques (factoring, conjugates, etc.) come in.

Constant Functions

If there’s no variable, nothing changes.

$$\lim_{x \to 7} 12 = 12$$

The limit of a constant is just the constant.

One-Sided Limits and Algebra

A one-sided limit means approach from only one direction:

• $\lim_{x \to c^-} f(x)$ from the left
• $\lim_{x \to c^+} f(x)$ from the right

If the expression is purely algebraic and defined at $c$, substitution gives the same value from both sides.

You need to think more carefully when:

• The function is piecewise
• The denominator becomes zero
• There’s behavior that depends on direction

Here’s a simple visual reminder of what left and right mean. In the graph, $x$ approaches 7 from both sides, and $f(x)$ approaches 8 even though there is an open circle at that point.

Left- and right-hand limits approaching the same value

A two-sided limit exists only if the left-hand and right-hand limits are equal. If they don’t match, the limit is DNE.

On FRQs, if they ask for a one-sided limit, make sure you’re using the correct piece of a piecewise function.

Limits of Composite Functions

Limits also work inside other functions.

If

$$\lim_{x \to c} g(x) = M$$

and the outside function is continuous at $M$, then

$$\lim_{x \to c} f(g(x)) = f(M).$$

In practice:

1. Find the inside limit.
2. Plug that result into the outside function.

Example:

$$\lim_{x \to 4} \sqrt{2x - 3}$$

First evaluate inside:

$$2(4) - 3 = 5$$

Then apply the root:

$$\sqrt{5}$$

Same idea with exponents:

$$\lim_{x \to 0} 3^{x^2}$$

Inside limit: $x^2 \to 0$.
So the whole limit becomes $3^0 = 1$.

The structure is always “inside first, then outside.”

Common Mistakes That Cost Points

• Using the quotient rule when the denominator’s limit is 0. The rule requires the denominator limit to be nonzero.
• Raising to a power before evaluating the inside limit. Evaluate the inside first.
• Forgetting that a two-sided limit requires matching one-sided limits.
• Doing extra algebra when substitution works immediately.

On multiple-choice, these often show up as tempting wrong answers. If plugging in works cleanly, trust it.