Limits and Their Properties

Average vs. Instantaneous Rate of Change

The average rate of change of a function over an interval measures the change in per unit change in across the interval. The instantaneous rate of change at a point is the derivative, representing how changes at a specific value of .

• Average Rate of Change:

• Instantaneous Rate of Change:

• Example: For , average rate from to is ; instantaneous rate at is .

Calculating Limits: First Steps and Indeterminate Forms

When calculating limits, the first step is to substitute the value into the function. If this leads to an indeterminate form (such as or ), further algebraic manipulation or limit laws are needed.

• Indeterminate Forms: , , , , , ,

• Example: yields ; factor numerator to get .

Techniques for Evaluating Limits

• Direct Substitution: Substitute the value directly if the function is continuous at that point.

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

• Rationalization: Multiply by the conjugate to simplify expressions with roots.

• Trigonometric Limits: Use known limits such as .

• Example:

Asymptotes and Graphing

Vertical and Horizontal Asymptotes

An asymptote is a line that a graph approaches but never touches. Vertical asymptotes occur where the function grows without bound as approaches a certain value. Horizontal asymptotes describe the behavior of a function as approaches infinity.

• Vertical Asymptote: If , then is a vertical asymptote.

• Horizontal Asymptote: If , then is a horizontal asymptote.

• Example: has a vertical asymptote at and a horizontal asymptote at .

Sketching Graphs and Identifying Asymptotes

• Identify points where the denominator is zero for vertical asymptotes.

• Compare degrees of numerator and denominator for horizontal asymptotes.

• For rational functions :

  • If degree of < degree of , is a horizontal asymptote.

  • If degrees are equal, .

  • If degree of > degree of , no horizontal asymptote.

Continuity and Types of Discontinuity

Definition of Continuity

A function is continuous at if:

• is defined

• exists

Types of Discontinuity

• Removable Discontinuity: Occurs when exists but is not defined or not equal to the limit.

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

• Infinite Discontinuity: The function approaches infinity at .

• Example: at has a removable discontinuity.

The Squeeze Theorem

Statement and Application

The Squeeze Theorem is used to find limits of functions trapped between two other functions whose limits are known.

• If for all near , and , then .

• Example:

The Intermediate Value Theorem (IVT)

Statement and Use

The Intermediate Value Theorem states that if is continuous on and is any number between and , then there exists in such that .

• Used to show the existence of roots or solutions within an interval.

• Example: If and , then must be zero for some in .

Introducing the Derivative

Definition and Geometric Interpretation

The derivative of at is the instantaneous rate of change of at , and is the slope of the tangent line to the graph at that point.

• Definition:

• Geometric Meaning: Slope of the tangent line to at

• Example: For ,

Equation of a Tangent Line

• The equation for the tangent line at is

• Example: For at , , , so

Derivative and Tangent Line Slope

• The derivative at a point gives the slope of the tangent line at that point.

• If does not exist, the graph is not differentiable at .

• Example: is not differentiable at .

Factorization and Graphing Techniques

Factoring Polynomials

• Factor expressions to simplify limits and find roots.

• Example:

Graphing Non-Differentiable Functions

• Points where the derivative does not exist may be cusps, corners, or vertical tangents.

• Example: has a corner at .

Summary Table: Types of Discontinuity

Practice and Preparation

• Review exercises from textbook sections 2.1–2.6 and 3.1–3.5.

• Focus on understanding concepts, not just memorizing procedures.

• Practice sketching graphs, identifying asymptotes, and calculating limits and derivatives.

Additional info: Some context and examples have been inferred and expanded for clarity and completeness based on standard Calculus I curriculum.