Limits and Continuity: Graphical and Numerical Approaches in Business Calculus

Limits and Continuity in Business Calculus

Introduction

Understanding limits and continuity is fundamental in Business Calculus, as these concepts underpin the study of rates of change, optimization, and modeling real-world business scenarios. This guide covers graphical and numerical approaches to evaluating limits, one-sided limits, and continuity, with examples and applications relevant to business contexts.

Limits: Definitions and Evaluation

Definition of a Limit

The limit of a function $f(x)$ as $x$ approaches a value $a$ is the value that $f(x)$ gets closer to as $x$ gets closer to $a$. Formally,

• Limit Notation: $\lim_{x \to a} f(x)$

• One-sided Limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right)

Limits can be evaluated numerically, graphically, or algebraically.

Evaluating Limits Graphically

• Observe the behavior of the function as $x$ approaches the target value from both sides.

• If the function approaches the same value from both sides, the two-sided limit exists.

• If the function approaches different values from the left and right, the two-sided limit does not exist, but one-sided limits may exist.

Example: For a piecewise function $g(x)$ defined as $g(x) = \frac{x+2}{2}$ for $x < 2$ and $g(x) = \frac{x}{2} + 1$ for $x \geq 2$, the graph shows a jump at $x=2$. The left and right limits at $x=2$ are:

• $\lim_{x \to 2^-} g(x) = 2$

• $\lim_{x \to 2^+} g(x) = 2$

• $\lim_{x \to 2} g(x) = 2$

Evaluating Limits Numerically

• Substitute values of $x$ close to the target value and observe the output.

• For rational functions, factor and simplify if direct substitution leads to indeterminate forms (e.g., $0/0$).

Example: $\lim_{x \to 5} \frac{x^2 - 9}{5 - x}$ Factor numerator: $x^2 - 9 = (x-3)(x+3)$ Substitute $x=5$: $\frac{25-9}{0} = \frac{16}{0}$ (undefined, so check one-sided limits or factor further if possible).

One-Sided Limits and Discontinuities

One-Sided Limits

One-sided limits are used when the function behaves differently from the left and right of a point.

• $\lim_{x \to a^-} f(x)$: Approach $a$ from values less than $a$.

• $\lim_{x \to a^+} f(x)$: Approach $a$ from values greater than $a$.

Example: For a function with a jump at $x=2$, the left and right limits may differ, indicating a discontinuity.

Types of Discontinuities

• Jump Discontinuity: The left and right limits exist but are not equal.

• Removable Discontinuity: The limit exists, but the function is not defined at that point or is defined differently.

• Infinite Discontinuity: The function approaches infinity as $x$ approaches the point.

Continuity of Functions

Definition of Continuity

A function $f(x)$ is continuous at $x=a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $\lim_{x \to a} f(x) = f(a)$

If any of these conditions fail, the function is discontinuous at $x=a$.

Checking Continuity Graphically

• Look for breaks, jumps, or holes in the graph at the point of interest.

• If the graph is unbroken and the function value matches the limit, the function is continuous.

Example: If $k(x)$ has a hole at $x=0$ but the limit exists, $k(x)$ is not continuous at $x=0$.

Piecewise Functions and Limits

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals. Limits at the boundaries require checking both one-sided limits.

• For $f(x) = \frac{x^2}{x+2}$ for $x < 2$, and $f(x) = x+2$ for $x \geq 2$, check $\lim_{x \to 2^-} f(x)$ and $\lim_{x \to 2^+} f(x)$.

Example: If both one-sided limits are equal, the two-sided limit exists.

Algebraic Techniques for Limits

Factoring and Simplifying

• Factor numerator and denominator to cancel common terms.

• Use direct substitution if the function is continuous at the point.

Examples:

• $\lim_{x \to 10} \frac{x^2 - 100}{x - 10}$ Factor numerator: $x^2 - 100 = (x-10)(x+10)$ Cancel $(x-10)$: $\lim_{x \to 10} x+10 = 20$

• $\lim_{x \to 5} \frac{x^2 + 3x - 40}{x^2 - 25}$ Factor numerator: $(x+8)(x-5)$, denominator: $(x+5)(x-5)$ Cancel $(x-5)$: $\lim_{x \to 5} \frac{x+8}{x+5} = \frac{13}{10}$

Summary Table: Types of Discontinuities

Type	Description	Graphical Feature
Jump	Left and right limits differ	Step/jump in graph
Removable	Limit exists, function value missing or different	Hole in graph
Infinite	Function approaches infinity	Vertical asymptote

Applications in Business Calculus

• Limits are used to define derivatives, which measure rates of change in cost, revenue, and profit functions.

• Continuity ensures that models behave predictably, which is essential for optimization and forecasting.

Practice Problems (from the file)

• Evaluate limits graphically and numerically for piecewise and polynomial functions.

• Determine continuity at specific points using graphs and algebraic definitions.

• Apply factoring and simplification to resolve indeterminate forms.

Key Formulas

• Limit Definition: $\lim_{x \to a} f(x)$

• Continuity at a Point: $\lim_{x \to a} f(x) = f(a)$

• Factoring for Limits: $\lim_{x \to a} \frac{(x-a)g(x)}{(x-a)h(x)} = \lim_{x \to a} \frac{g(x)}{h(x)}$ (if $h(a) \neq 0$)

Additional info: The study notes expand on graphical and numerical limit evaluation, continuity, and piecewise functions, as presented in the original questions. These topics are foundational for Business Calculus, especially in modeling and analyzing business functions.