Algebraic Limits and Continuity

Direct Substitution and Rational Functions

For any rational function $f$, with $c$ in the domain of $f$, the limit as $x$ approaches $c$ is equal to the function evaluated at $c$:

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

Polynomial functions and rational functions where $c$ is within the domain allow for direct evaluation. When evaluating limits, direct substitution is always the first approach. If direct substitution yields an undefined expression (such as an indeterminate form $\frac{0}{0}$), the algebraic expression must be rewritten—by factoring, simplifying, splitting, or combining terms—and evaluated again by substitution.

Fundamental Rules of Limits

The fundamental algebraic properties and rules for limits of functions, where $c$ is within the domain, are defined as follows:

The constant rule:

$\lim_{x \to c} c = c$

The sum and difference rule:

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

The product rule:

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

The quotient rule:

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

This quotient rule is valid provided that the denominator evaluates to a non-zero value, meaning $g(c) \neq 0$

The power rule:

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

Graphical Behavior: Holes versus Asymptotes

When working with rational functions where $c$ is not in the domain, two primary graphical situations arise when division by zero occurs:

A hole (removable discontinuity) occurs when a factor in the denominator cancels out with a factor in the numerator. Although $c$ is not in the domain of the function, the limit as $x$ approaches $c$ exists. Graphically, as $x \to c$, the curve approaches a specific finite value from both the left and the right sides.

An asymptote (vertical asymptote) occurs when a zero in the denominator does not cancel out. In this case, $c$ is not in the domain of the rational function, and as $x$ approaches $c$, the function values expand toward positive infinity ($+\infty$) or negative infinity ($-\infty$). Consequently, a finite two-sided limit does not exist at a vertical asymptote.

Worked Examples of Limit Evaluation

Example 5 demonstrates limit evaluation for a quadratic polynomial function using direct substitution:

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

Since $x^2 + 3x - 5$ is a polynomial function, direct substitution at $x = 2$ yields $f(2)$:

$f(2) = (2)^2 + 3(2) - 5$

$f(2) = 4 + 6 - 5$

$f(2) = 5$

Therefore, the limit value is:

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

Example 6 demonstrates evaluating a limit that initially yields an undefined expression:

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

Direct substitution of $x = 2$ gives $\frac{0}{0}$, which is undefined. Factoring the numerator as a difference of squares gives:

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

Canceling the common factor $(x - 2)$ indicates the presence of a hole at $x = 2$, meaning the limit exists. The simplified expression becomes:

$\lim_{x \to 2} (x + 2)$

Evaluating by direct substitution gives:

$2 + 2 = 4$

Therefore, the final limit value is:

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