Limits and Continuity

Introduction to Limits

Limits are foundational to calculus, describing the behavior of functions as inputs approach specific values. Understanding limits is essential for defining derivatives and integrals.

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

• Notation:

• Indeterminate Forms: Expressions like or require algebraic manipulation to resolve.

Example: can be simplified by factoring numerator and denominator, yielding .

• Laws of Limits:

  • (if denominator ≠ 0)

Special Limit: (important in trigonometric limits).

Differentiation

Differentiation from First Principles

Differentiation measures how a function changes as its input changes. The derivative at a point is the slope of the tangent to the curve at that point.

• Definition: The derivative of f(x) at x is

• Example: For , the derivative is found by expanding , subtracting , dividing by h, and taking the limit as h approaches 0.

Rules of Differentiation

• Power Rule:

• Constant Rule:

• Sum Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Example (Product Rule): If , then .

Differentiating Trigonometric, Exponential, and Logarithmic Functions

• Trigonometric:

• Exponential:

• Logarithmic:

Example:

Applications of Differentiation

Slope of a Tangent to a Curve

The derivative at a point gives the slope of the tangent to the curve at that point.

• Formula: Slope at is

• Example: For , at , slope is

Finding Maxima, Minima, and Points of Inflection

Critical points occur where the derivative is zero or undefined. The second derivative test helps classify these points.

• Maxima/Minima: Occur where

• Second Derivative Test:

  • If , point is a minimum

  • If , point is a maximum

  • If , test is inconclusive (possible inflection point)

Rates of Change

Derivatives represent rates of change in various contexts, such as velocity (rate of change of position) and acceleration (rate of change of velocity).

• Velocity: , where is position

• Acceleration:

Summary Table: Differentiation Rules

Additional info:

• These notes are based on the Leaving Certificate Higher Level Calculus syllabus, covering limits, differentiation, and their applications.

• Further topics such as curve sketching, implicit differentiation, and applications to real-world problems are also included in the full set of notes.