Limits and Continuity

Introduction to Limits

Limits are a foundational concept in calculus, describing the behavior of functions as inputs approach specific values. Understanding limits is essential for defining derivatives, continuity, and analyzing function behavior near points of interest.

• 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:

• Key Properties:

  • If the left-hand and right-hand limits are equal, the limit exists.

  • Limits can be finite or infinite, and may not exist if the function behaves erratically near the point.

• Example:

Evaluating Limits

Limits can be evaluated using direct substitution, factoring, rationalization, or special limit laws. Some limits require recognizing indeterminate forms and applying algebraic techniques.

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

• Factoring: Used when substitution yields 0/0; factor numerator and denominator to simplify.

• Rationalization: Multiply by a conjugate to simplify expressions involving roots.

• Special Trigonometric Limits: and .

• Example:

  • Factor numerator:

  • Simplify:

Limits Involving Trigonometric Functions

Trigonometric limits often require using identities or recognizing standard forms.

• Example:

  • Evaluate: ,

  • Result:

• Example:

  • Use:

  • Result: $5$

Limits Involving Roots and Rational Functions

Limits with roots or rational expressions may require algebraic manipulation.

• Example:

  • Direct substitution yields 0/0; rationalize numerator.

  • Multiply numerator and denominator by .

  • Simplify and evaluate the limit.

Continuity

A function is continuous at a point if its limit exists at that point and equals the function's value there. Continuity is crucial for applying calculus concepts like differentiation and integration.

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

• Interval Notation: Used to describe where a function is continuous.

• Types of Discontinuity:

  • Removable: The limit exists, but f(a) is undefined or not equal to the limit.

  • Non-removable: The limit does not exist due to jump or infinite discontinuity.

• Example: is discontinuous at x = 1, but the discontinuity is removable.

Average and Instantaneous Rate of Change

Average Speed

The average rate of change of a function over an interval is the change in output divided by the change in input.

• Formula:

• Application: For a skydiver with position , average speed between and is .

Instantaneous Speed (Derivative)

The instantaneous rate of change at a point is the derivative, found using the limit of the difference quotient.

• Formula:

• Application: For , the instantaneous speed at is .

Tangent Line to a Curve

The tangent line to a function at a point has a slope equal to the derivative at that point and passes through the point.

• Equation:

• Example: For at , the tangent line is .

Solving Equations and the Intermediate Value Theorem

Intermediate Value Theorem (IVT)

The IVT states that if a function is continuous on [a, b] and takes values f(a) and f(b), then it takes every value between f(a) and f(b) at some point in (a, b).

• Application: To show that has a root in [2, 3], check that and have opposite signs.

Summary Table: Types of Discontinuity

Key Formulas and Theorems

• Limit Definition of Derivative:

• Average Rate of Change:

• Equation of Tangent Line:

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

Practice Problems (from Exam)

• Compute limits involving trigonometric, rational, and root functions.

• Determine intervals of continuity and classify discontinuities.

• Apply the IVT to show existence of roots.

• Find average and instantaneous rates of change for position functions.

• Write equations of tangent lines at specified points.

Additional info: The above guide expands on the exam questions, providing definitions, formulas, and context for each topic covered. Students should practice evaluating limits, identifying continuity, and applying theorems as shown.