Limits and Continuity

Introduction

This study guide covers fundamental concepts in limits and continuity, as presented in a typical Calculus I problem set. Understanding limits is essential for analyzing the behavior of functions near specific points and forms the foundation for the study of derivatives and integrals.

Evaluating Limits

Definition of a Limit

• Limit: The value that a function f(x) approaches as x approaches a particular point a.

• Notation: means that as x gets arbitrarily close to a, f(x) gets arbitrarily close to L.

Evaluating Limits Algebraically

• Direct Substitution: If f(x) is continuous at a, then .

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

• Rationalization: Multiply by a conjugate to simplify expressions involving roots.

• Special Limits: Recognize standard limits such as .

One-Sided Limits

• Left-hand limit: is the value as x approaches a from the left.

• Right-hand limit: is the value as x approaches a from the right.

• The two-sided limit exists only if both one-sided limits exist and are equal.

Limits Involving Piecewise Functions

• For piecewise functions, evaluate the limit from each side using the appropriate formula for f(x).

• If the left- and right-hand limits are not equal, the limit does not exist at that point.

Examples

• Example 1: Factor numerator: Simplify: (for ) Substitute: Answer: 4

• Example 2: (standard limit)

Determining When Limits Do Not Exist

• Limits may not exist if:

  • The left- and right-hand limits are not equal.

  • The function grows without bound (infinite limit).

  • The function oscillates as x approaches a.

Limits Involving Parameters

Finding Values for Existence of Limits

• Sometimes, a limit will exist only for certain values of a parameter (e.g., in ).

• Set the numerator to zero at the problematic point to ensure the limit exists and solve for the parameter.

Continuity and Removable Discontinuities

Definition of Continuity

• Continuous at a point: f(x) is continuous at x = a if:

  • is defined

  • exists

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

Special Trigonometric and Radical Limits

• Use trigonometric identities and standard limits for expressions involving , , etc.

• For radicals, rationalize the numerator or denominator as needed.

Limits at Infinity and Infinite Limits

• As x approaches infinity, compare the degrees of numerator and denominator in rational functions:

  • If degrees are equal: limit is the ratio of leading coefficients.

  • If numerator degree < denominator degree: limit is 0.

  • If numerator degree > denominator degree: limit does not exist (infinite).

Average and Instantaneous Velocity

Definitions

• Average velocity: , where is the position function.

• Instantaneous velocity: The derivative , or .

Example

• Given , find average velocity from to :

• Compute and , then use the average velocity formula.

• For instantaneous velocity at , compute and evaluate at .

Sample Table: Types of Discontinuities

Summary of Key Formulas

• Limit definition:

• Average velocity:

• Instantaneous velocity:

• Standard trigonometric limits:

Practice Problems

• Evaluate limits using algebraic simplification, factoring, and rationalization.

• Determine values of parameters for which limits exist.

• Analyze piecewise functions for continuity and removable discontinuities.

• Apply limit definitions to velocity problems.