Limits and Limit Notation

Understanding Limits

Limits describe the behavior of a function as the input approaches a particular value. They are foundational in calculus for defining continuity, derivatives, and integrals.

• Limit Notation: represents the value that f(x) approaches as x approaches c.

• Left-hand limit: (approaching from values less than c).

• Right-hand limit: (approaching from values greater than c).

• Existence of Limit: exists if and only if the left and right limits are equal.

• Undefined Limits: If the left and right limits are not equal, the limit does not exist (DNE).

Example: For with a jump at , , so DNE.

Evaluating Limits Graphically and Algebraically

• Use the graph to determine the value f(x) approaches from both sides of c.

• For polynomials, plug in the value directly since they are continuous everywhere.

• For rational functions, check if the denominator becomes zero; if so, use factoring or conjugate multiplication to simplify.

Example:

Special Limit Cases and Techniques

Indeterminate Forms and Factoring

When direct substitution yields , the form is indeterminate. Use algebraic manipulation:

• Factoring: Factor numerator and denominator to cancel common terms.

• Conjugate Multiplication: Multiply by the conjugate to simplify expressions involving square roots.

Example: Factor numerator: Cancel :

Limits Involving Infinity and Rational Functions

Limits as x approaches infinity help determine horizontal asymptotes.

• If degree of numerator < degree of denominator:

• If degrees are equal:

• If degree of numerator > degree of denominator: Limit is or

Example:

Using Tables to Estimate Limits

When substitution is not possible, use a table of values approaching c from both sides to estimate the limit.

Interpretation: As x approaches 1 from the left, ; from the right, .

Continuity

Definition of Continuity at a Point

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

• 1. f(c) is defined

• 2. exists

• 3.

If any of these conditions fail, the function is not continuous at c.

Example: For with a hole at , is undefined, so is not continuous at .

Types of Discontinuity

• Removable Discontinuity: A hole in the graph; limit exists but is undefined or not equal to the limit.

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

• Infinite Discontinuity: Function approaches infinity at c.

Slope, Secant, and Tangent Lines

Secant Line

The secant line connects two points on a curve and its slope represents the average rate of change between those points.

• Slope of Secant Line:

Tangent Line and Instantaneous Rate of Change

The tangent line touches the curve at one point and its slope represents the instantaneous rate of change (the derivative).

• Slope of Tangent Line:

Definition of the Derivative

The derivative of f(x) at x is defined as:

Interpretation: The derivative gives the slope of the tangent line at any point x.

Derivative Rules and Computation

Basic Derivative Rules

• Constant Rule:

• Line Rule:

• Power Rule:

Examples of Derivatives

• Example 1:

• Example 2:

• Example 3:

Algebraic Manipulation Before Differentiation

• Rewrite roots and reciprocals as powers before applying the power rule.

• Example:

Applications: Profit, Revenue, and Cost Functions

Definitions

• Profit Function: , where R(x) is revenue and C(x) is cost.

• Revenue: , where p is price and x is units sold.

• Marginal Cost/Revenue/Profit: The derivative of the cost, revenue, or profit function, representing the rate of change with respect to units produced or sold.

Example: If , then marginal cost is .

Vertical and Horizontal Asymptotes

Horizontal Asymptote

Determined by the limit as x approaches infinity:

Vertical Asymptote

Occurs where the denominator of a rational function is zero and the numerator is not zero.

• Find x values that make denominator zero.

• Check if numerator is also zero (hole) or not (vertical asymptote).

Summary Table: Types of Discontinuity

Additional info:

• Some exercises require using tables to estimate limits, especially for indeterminate forms.

• Graphical analysis is important for understanding continuity and differentiability.

• Applied problems include cost, revenue, and profit functions, emphasizing the use of derivatives for marginal analysis.