Limits and Continuity

Introduction

This study guide covers the foundational concepts of limits and continuity in calculus, as presented in a practice exam format. Understanding limits is essential for grasping the behavior of functions near specific points and forms the basis for further topics such as derivatives and integrals.

Fundamentals of Limits

Limits describe the value that a function approaches as the input approaches a certain point. They are crucial for analyzing function behavior, especially at points of discontinuity or where direct evaluation is not possible.

• Definition of a Limit: The limit of a function f(x) as x approaches a is denoted as .

• Existence of Limits: A limit exists if the left-hand and right-hand limits are equal as x approaches a.

• One-Sided Limits: (from the left), (from the right).

• Infinite Limits: If f(x) increases or decreases without bound as x approaches a, the limit is infinite.

• Limits at Infinity: describes the behavior of f(x) as x becomes very large.

Evaluating Limits Algebraically

Limits can often be evaluated by direct substitution, factoring, rationalizing, or applying limit laws.

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

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

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

• Limit Laws:

  • , provided

Examples of Limit Problems

• Polynomial and Rational Functions:

• Square Roots and Conjugates:

• Indeterminate Forms: When direct substitution yields or , use algebraic manipulation or L'Hospital's Rule (if allowed).

• Piecewise and Graphical Limits: Use the graph to determine left and right limits, and check for discontinuities or holes.

Limits from Graphs

Limits can be estimated or determined from the graph of a function by observing the behavior as x approaches a specific value.

• Open and Closed Dots: An open dot indicates the function is not defined at that point; a closed dot shows the actual value.

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

• Example: Given a graph, find , , , and .

Continuity

A function is continuous at a point a if:

• is defined

• exists

Discontinuities occur when any of these conditions fail.

Special Limit Techniques

• Limits Involving Square Roots: Rationalize the numerator or denominator to simplify.

• Limits at Infinity: For rational functions, compare degrees of numerator and denominator.

• Limits with Piecewise Functions: Evaluate each piece separately and check for agreement at the boundary.

Sample Table: Limit Evaluation Methods

Practice Problems (from Exam)

• Evaluate given , .

• Evaluate given , .

• Given a graph, find , , , , .

• Evaluate and .

• Evaluate and .

• Evaluate and .

• Evaluate and .

Summary Table: Types of Discontinuities

Key Terms

• Limit

• Continuity

• Indeterminate Form

• Removable Discontinuity

• Jump Discontinuity

• Infinite Discontinuity

Additional info:

• Some problems may require graphical analysis or interpretation of piecewise functions.

• Practice problems are representative of typical Calculus I exam questions on limits and continuity.