Limits, Continuity, and Derivatives

Section 2.3: Evaluating Limits

Understanding limits is foundational in calculus, as they describe the behavior of functions as inputs approach specific values. This section covers methods for finding limits using algebraic rules and the Squeeze Theorem.

• Limit Laws: These are algebraic properties that allow the computation of limits for sums, products, quotients, and compositions of functions. Common laws include:

  • Sum Law:

  • Product Law:

  • Quotient Law: , provided

• Squeeze Theorem: If for all near , and , then .

• Example: To find , use the Squeeze Theorem since implies and both bounds approach 0 as .

Section 2.4: Infinite Limits and Vertical Asymptotes

Infinite limits describe the behavior of functions that increase or decrease without bound near certain points, often leading to vertical asymptotes.

• Vertical Asymptote: A line is a vertical asymptote if or .

• Infinite Limits: If grows arbitrarily large as approaches , we write or .

• Example: ; thus, is a vertical asymptote for .

Section 2.5: Limits at Infinity and Horizontal Asymptotes

Limits at infinity help determine the end behavior of functions and the existence of horizontal asymptotes.

• Horizontal Asymptote: The line is a horizontal asymptote if or .

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

  • If degrees are equal:

  • If numerator degree < denominator: limit is 0

  • If numerator degree > denominator: limit is or

• Example:

Section 2.6: Continuity and the Intermediate Value Theorem

Continuity ensures that a function has no breaks, jumps, or holes at a point. The Intermediate Value Theorem (IVT) is a key result for continuous functions.

• Continuity Checklist at :

  1. is defined

  2. exists

• Intermediate Value Theorem (IVT): If is continuous on and is between and , then there exists such that .

• Application: IVT is used to show the existence of roots or solutions within an interval.

• Example: If and , then for some in .

Section 2.7: Formal Definitions of Limits

This section introduces the rigorous (epsilon-delta) definition of a limit and the formal definition of infinite limits.

• Epsilon-Delta Definition: means that for every , there exists such that if , then .

• Infinite Limit Definition: means that for every , there exists such that if , then .

• Graphical Interpretation: Given , find so that the function stays within of whenever is within of .

• Example: Prove using the epsilon-delta definition.

Section 3.1: The Derivative and Tangent Lines

The derivative measures the instantaneous rate of change of a function and is geometrically interpreted as the slope of the tangent line at a point.

• Definition of the Derivative at a Point:

• Equation of the Tangent Line:

• Example: For , the tangent line at is

Section 3.2: Calculating Derivatives and Analyzing Graphs

This section focuses on finding derivatives using the limit definition, sketching derivatives, and identifying points of discontinuity and non-differentiability.

• Limit of the Difference Quotient:

• Finding Tangent Lines: Use the derivative at a point to write the equation of the tangent line.

• Sketching from : The slope of at each point gives the value of at that point.

• Continuity and Differentiability:

  • Not Continuous: Points where the function has jumps, holes, or vertical asymptotes.

  • Not Differentiable: Points where the function is not continuous, has sharp corners (cusps), or vertical tangents.

• Example: The function is not differentiable at due to a sharp corner.

Additional info: This guide expands on the listed exam topics with definitions, examples, and a summary table for continuity and differentiability.