12.2 Finding Limits: Properties of Limits

Properties of Limits

Limits are the foundation of calculus, allowing you to analyze function behavior near specific points. They help you understand continuity, rates of change, and what happens to function values at tricky spots.

The properties of limits simplify complex calculations by breaking them into manageable parts. These rules cover basic operations, polynomials, powers, roots, and rational functions, plus how to handle indeterminate forms like $\frac{0}{0}$.

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Properties of Limits

Limits of basic algebraic operations

All of these properties assume that $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ both exist. If either limit doesn't exist, you can't apply these rules directly.

• Sum Rule: The limit of a sum equals the sum of the limits.

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

• Difference Rule: The limit of a difference equals the difference of the limits.

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

• Product Rule: The limit of a product equals the product of the limits.

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

• Constant Multiple Rule: You can pull a constant out of a limit.

$\lim_{x \to a} (c \cdot f(x)) = c \cdot \lim_{x \to a} f(x)$ where $c$ is a constant.

These rules let you break a complicated limit into smaller pieces, evaluate each piece, and then combine the results. For example, to find $\lim_{x \to 3} (2x + x^2)$, you can find $\lim_{x \to 3} 2x$ and $\lim_{x \to 3} x^2$ separately, then add them: $6 + 9 = 15$.

Limits of polynomial functions

Polynomial functions are continuous everywhere, which means the limit as $x$ approaches $a$ is just the function value at $a$. This gives you the simplest possible method: direct substitution.

To find $\lim_{x \to a} P(x)$ where $P(x)$ is a polynomial, just plug $a$ into the polynomial.

Example: For $P(x) = 3x^2 - 2x + 1$:

$\lim_{x \to 2} P(x) = P(2) = 3(2)^2 - 2(2) + 1 = 12 - 4 + 1 = 9$

No special techniques needed. Direct substitution works every time for polynomials because they have no holes, jumps, or asymptotes.

Limits with powers and roots

• Power Rule: The limit of a power equals the power of the limit.

$\lim_{x \to a} (f(x))^n = \left(\lim_{x \to a} f(x)\right)^n$ where $n$ is a real number.

• Root Rule: The limit of an $n$th root equals the $n$th root of the limit.

$\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$ where $n$ is a positive integer. For even roots, the limit of the radicand must be non-negative.

• Exponential Function Rule: The limit of an exponential function equals the base raised to the limit of the exponent.

$\lim_{x \to a} b^{f(x)} = b^{\lim_{x \to a} f(x)}$ where $b > 0$ and $b \neq 1$.

The pattern across all three rules is the same: evaluate the limit of the "inner" function first, then apply the outer operation (power, root, or exponential) to the result.

Limits of rational functions

Quotient Rule: The limit of a quotient equals the quotient of the limits, provided the denominator's limit isn't zero.

$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad \text{where } \lim_{x \to a} g(x) \neq 0$

When the denominator's limit isn't zero, just use direct substitution. The interesting cases are when it is zero.

Indeterminate Form $\frac{0}{0}$

This occurs when both the numerator and denominator approach 0. The limit might still exist, but you need to do more work to find it. The most common strategy is factoring and canceling:

1. Factor the numerator and denominator.
2. Cancel the common factor that's causing both to equal zero.
3. Evaluate the simplified expression using direct substitution.

Example:

$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$

Direct substitution gives $\frac{0}{0}$, so factor instead:

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

Indeterminate Form $\frac{\infty}{\infty}$

This occurs when both the numerator and denominator grow without bound (typically as $x \to \infty$). To resolve it, divide every term in the numerator and denominator by the highest power of $x$ present.

Example:

$\lim_{x \to \infty} \frac{3x^2 + 2x}{5x^2 - 1}$

Divide everything by $x^2$:

$= \lim_{x \to \infty} \frac{3 + \frac{2}{x}}{5 - \frac{1}{x^2}} = \frac{3 + 0}{5 - 0} = \frac{3}{5}$

The terms with $x$ in the denominator go to 0 as $x \to \infty$, leaving only the leading coefficients.

Continuity and Discontinuities

A function $f(x)$ is continuous at $x = a$ if three conditions are all met:

1. $f(a)$ is defined.
2. $\lim_{x \to a} f(x)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$.

If any of these fails, the function has a discontinuity at that point. Discontinuities come in a few types:

• Removable discontinuity: The limit exists, but the function is either undefined or has a different value at that point. On a graph, this looks like a hole. You could "fill it in" by redefining the function at that single point.
• Jump discontinuity: The left-hand and right-hand limits both exist but aren't equal. The graph has a sudden jump.
• Infinite discontinuity: The function approaches $\infty$ or $-\infty$ near the point, producing a vertical asymptote.

Why does continuity matter for limits? When a function is continuous at $a$, you can always use direct substitution. Discontinuities are exactly the spots where you'll need the algebraic techniques from the rational functions section above.