Limit Basics

Definition and Interpretation of Limits

The concept of a limit is foundational in calculus, describing the behavior of a function as its input approaches a particular value.

• Limit Notation: means that as approaches , approaches .

• Key Points:

  • The limit describes what approaches as gets close to , not necessarily the value at .

  • does not guarantee that ; it only describes the behavior near .

• One-Sided Limits: Be familiar with left-hand () and right-hand () limits.

• Limit Existence: only if both one-sided limits exist and are equal to .

• Graphical Interpretation: Be able to use a graph to answer questions about limits.

Examples

• Example: For , as , approaches $2f(1)$ is undefined.

Continuity

Definition and Properties

Continuity describes a function that does not have any abrupt jumps, holes, or breaks at a point or over an interval.

• Definition: is continuous at if .

• One-Sided Continuity: There is one-sided continuity if the one-sided limit equals the function value.

• Interval Continuity: is continuous on an interval if it is continuous at every point in the interval.

• Common Points of Discontinuity:

  • Division by zero

  • Even roots of negative numbers

  • Logarithms of non-positive numbers

  • Piecewise rule changes or undefined points

Examples

• Example: is discontinuous at due to division by zero.

Limit Evaluation Techniques

Algebraic Methods

Limits can often be evaluated using algebraic manipulation, especially when direct substitution leads to indeterminate forms.

• Factoring: If the limit is of the form , try to factor numerator and denominator and simplify.

• Multiplying by Conjugate: For limits involving square roots, multiply by the conjugate to simplify.

• Handling Fractions: Know how to handle limits that involve complex fractions.

• One-Sided Limits: For forms like , consider the direction of approach and sign.

• Notation: Be careful not to drop the limit operator until the operation is complete.

Examples

• Example: : Factor numerator to get , so the limit is $4$.

Limit Definition of the Derivative

Concept and Application

The derivative of a function at a point measures the instantaneous rate of change, defined as the limit of the difference quotient.

• Definition:

• Geometric Interpretation: The derivative gives the slope of the tangent line to at .

• Secant and Tangent Lines: The slope of the secant line approximates the average rate of change; the tangent line gives the instantaneous rate.

• Equation of Tangent Line:

Examples

• Example: For , .

Derivative Rules

Basic Derivative Formulas

Knowing the derivatives of basic functions and how to apply rules is essential for differentiation.

• Constant Rule:

• Power Rule:

• Exponential Rule:

• Logarithmic Rule:

• Trigonometric Rules:

• Inverse Trigonometric Rules:

• Sum Rule:

• Constant Multiple Rule:

• Product Rule:

• Quotient Rule:

Exponential and Logarithmic Rules

• Exponential Rule:

• Logarithmic Rules:

Implicit Differentiation

Concept and Application

Implicit differentiation is used when a function is not given explicitly as , but rather in a form involving both and .

• Solving for : Differentiate both sides of the equation with respect to , treating as a function of .

• Finding Slopes: gives the slope at all points on the curve; use it to find the slope at a specific point.

• Horizontal/Vertical Tangents: Set for horizontal tangents, and solve for vertical tangents where the denominator is zero.

• Logarithmic Differentiation: Use logarithmic differentiation for functions of the form .

• Know the log and exponential rules.

Example

• Example: For , differentiate both sides to get , so .

Derivative Table

Common Derivatives

The following table summarizes the derivatives of common functions and rules:

Exponential and Logarithmic Rules Table

Additional info:

• Some notation and examples have been expanded for clarity and completeness.

• Tables have been reconstructed to summarize key derivative and logarithmic rules.