Limits and Continuity

Evaluating Limits

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

• Definition: The limit of a function f(x) as x approaches a value a is the value that f(x) approaches as x gets arbitrarily close to a.

• Notation:

• Techniques:

  • Direct substitution

  • Factoring and simplifying

  • Rationalizing

  • Using special limits (e.g., )

• One-sided limits: (from the left), (from the right)

• Infinite limits: When f(x) increases or decreases without bound as x approaches a value.

• Limits at infinity: describes the end behavior of a function.

Example: Factor numerator and denominator, then simplify and substitute.

Continuity

A function is continuous at a point if the limit exists and equals the function value at that point.

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

• Types of discontinuities: removable, jump, infinite

Example: Piecewise functions may have discontinuities at the points where the formula changes.

Differentiation

Definition of the Derivative

The derivative of a function measures the rate at which the function value changes as its input changes. It is the foundation for understanding slopes, rates of change, and motion.

• Definition:

• Notation: , ,

Basic Differentiation Rules

• Power Rule:

• Constant Rule:

• Constant Multiple Rule:

• Sum Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Example:

Derivatives of Trigonometric, Exponential, and Logarithmic Functions

Example: (by chain rule)

Implicit Differentiation

Used when y is defined implicitly as a function of x (not solved for y explicitly).

• Differentiating both sides of an equation with respect to x, treating y as a function of x.

• Apply the chain rule to terms involving y.

Example: For ,

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to another, often with respect to time.

• Differentiate both sides of an equation with respect to time t.

• Use the chain rule for variables that are functions of t.

Example: If the radius of a circle increases at 0.2 m/s, the rate of change of area A is

Applications of Derivatives

Tangent Lines

The tangent line to a curve at a point is the straight line that just touches the curve at that point and has the same slope as the curve there.

• Equation: , where is the derivative at .

Example: Find the tangent to at : Slope is , so .

Horizontal Tangents

A tangent is horizontal where the derivative is zero.

• Solve to find x-values where the tangent is horizontal.

Motion Problems

Describes the position, velocity, and acceleration of objects moving along a line.

• Position function: s(t)

• Velocity:

• Acceleration:

• Maximum height occurs when

• Time to hit the ground: solve

Example: For ,

Graphical Analysis

Limits from Graphs

Limits can be estimated or found exactly by analyzing the behavior of a function's graph as x approaches a value.

• Look for the y-value the graph approaches from both sides.

• If the left and right limits differ, the limit does not exist.

Asymptotes

• Vertical asymptotes: Occur where the function grows without bound as x approaches a certain value (often where denominator is zero).

• Horizontal asymptotes: Describe end behavior as or .

Example: has vertical asymptotes where , i.e., .

Solving Equations and Quadratics

Quadratic Equations

• Standard form:

• Quadratic formula:

Example: Solve using the quadratic formula.

Solving Trigonometric Equations

• Find all solutions in a given interval, often using inverse trig functions.

• Example: Solve for .

Special Derivatives and Inverse Functions

• Inverse trigonometric derivatives:

• Logarithmic differentiation: Useful for functions of the form .

Summary Table: Key Derivative Rules

Additional info:

• This study guide covers the main topics from a Calculus I exam or homework set, including limits, derivatives, applications to motion, related rates, and graphical analysis.

• Some problems involve interpreting graphs, solving equations, and applying calculus to real-world scenarios.