Limits and Their Applications in Business Calculus

Limits

Definition of a Limit

The concept of a limit is foundational in calculus and describes the value that a function approaches as the input approaches a certain point.

• Limit Notation: $\lim_{x \to a} f(x) = L$ means as $x$ approaches $a$, $f(x)$ approaches $L$.

• One-Sided Limits:

  • Left-hand limit: $\lim_{x \to a^-} f(x)$

  • Right-hand limit: $\lim_{x \to a^+} f(x)$

• Important Note: The limit value at $x$ may or may not equal the function value at $x$.

Finding Limits and Function Values on a Graph

Limits can be estimated by analyzing the behavior of a function on its graph as $x$ approaches a specific value.

• Identify the value the function approaches from both sides of the point.

• If the left and right limits are equal, the two-sided limit exists.

• If the function jumps or has a hole, the limit may not exist or may differ from the function value.

Sketching Functions with Given Limits

Graphical Representation

Sketching functions based on specified limits and function values helps visualize continuity and discontinuity.

• Plot points for $f(a)$ and ensure the graph approaches the given limits from both sides.

• Discontinuities (holes, jumps) are indicated where the limit does not match the function value.

Limit Laws

Algebraic Properties of Limits

Limit laws allow the computation of limits for sums, products, quotients, and compositions of functions.

Law	Formula
Sum Law	$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
Difference Law	$\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$
Product Law	$\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
Quotient Law	$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, if $\lim_{x \to a} g(x) \neq 0$
Constant Multiple Law	$\lim_{x \to a} [c f(x)] = c \lim_{x \to a} f(x)$
Power Law	$\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$
Root Law	$\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$

Calculating Limits Given Algebraic Expressions

Direct Substitution Property

If $f$ is a polynomial or rational function and $a$ is in its domain, then:

• $\lim_{x \to a} f(x) = f(a)$

Other Methods

• Factor and simplify the expression.

• Use conjugates for expressions involving roots.

• Set up a table to observe the function's behavior as $x$ approaches the limit.

Infinity and Does Not Exist (DNE)

Limits at Infinity

When a function increases or decreases without bound as $x$ approaches a value, the limit is said to be infinity or negative infinity.

• If $f(x)$ grows larger as $x$ approaches $a$, $\lim_{x \to a} f(x) = \infty$.

• If $f(x)$ decreases without bound, $\lim_{x \to a} f(x) = -\infty$.

Asymptotes of Rational Functions

Vertical Asymptotes

A vertical asymptote occurs when the function approaches infinity as $x$ approaches a restricted value.

• Occurs at $x = a$ if $\lim_{x \to a^-} f(x) = \pm\infty$ or $\lim_{x \to a^+} f(x) = \pm\infty$.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as $x$ approaches infinity.

• If $\lim_{x \to \infty} f(x) = L$, then $y = L$ is a horizontal asymptote.

Limits as x Approaches Infinity

Evaluating Limits at Infinity

For rational functions, the degree of the numerator and denominator determines the limit as $x$ approaches infinity.

Case	Result
Degree of numerator < degree of denominator	Limit is 0
Degree of numerator = degree of denominator	Limit is ratio of leading coefficients
Degree of numerator > degree of denominator	Limit is infinity or negative infinity

Piecewise Functions and Limits

Evaluating Limits for Piecewise Functions

Identify the boundary from the left or right and use the appropriate function to evaluate the limit.

• Check which piece of the function applies as $x$ approaches the boundary.

• Calculate left-hand and right-hand limits separately if needed.

Summary Table: Limit Laws

Law	Formula
Sum	$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
Difference	$\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$
Product	$\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
Quotient	$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$
Constant Multiple	$\lim_{x \to a} [c f(x)] = c \lim_{x \to a} f(x)$
Power	$\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$
Root	$\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$

Examples

• Example 1: $\lim_{x \to 2} (3x + 1) = 7$

• Example 2: $\lim_{x \to 0} \frac{2x}{x^2 + 1} = 0$

• Example 3: For $f(x)$ defined piecewise, $\lim_{x \to 1^-} f(x)$ and $\lim_{x \to 1^+} f(x)$ may differ.

Additional info: These notes cover the foundational concept of limits, which is essential for understanding derivatives and integrals in Business Calculus. The material is directly relevant to Chapter 3 (The Derivative) and Chapter 4 (Calculating the Derivative), as limits are the basis for defining and computing derivatives.