calc notes

Continuous Functions and Limits

  • Continuity at a point

    • Example: F(2) = 2

    • Behavior of the function as x approaches 2

      • Approaching from either side gives limits that agree.

Polynomial Functions

  • General form: x squared is x squared

  • Finding limits: Limit as x approaches -1

    • Approach from either side results in values approaching 1.

    • If it exists, evaluate: plug in the value to find the function's output.

  • Simplifying Functions:

    • Example of combining functions: x² + x - 1 or x² - 5

    • Use polynomial extension for limits.

    • Important: Always divide if encountering a limit.

Evaluating Limits

  • First examples:

    • Limit when x approaches 0:

      • Calculations: ( 0^2 - 2 ) divided by ( 0^3 - 1 )

      • If the function has a discontinuity such as undefined points, more analysis needed.

    • Discontinuous Functions:

      • Defined in intervals, piecewise functions split at points like x=1.

      • Behavior changes at points of discontinuity; evaluate from both sides.

  • Right-hand limit and left-hand limit definitions:

    • When x approaches 1 from the left (denoted as 1^-), use the function from left piece.

    • Value is found by plugging x=1 into Left hand side pieces.

Analysis of Function Behavior

  • Aggregate results: limits can differ from sides.

    • Example: Define if function f(x) is continuous; if output differs drastically when approaching limits.

    • Assess if f(1) exists; check for function definition to confirm.

Continuity and Polynomial Behavior

  • Polynomial limits:

    • If function is polynomial, limits can be evaluated directly by plugging in values.

  • Rational Functions:

    • Case 1: q(a) != 0 (non-zero case) results in simply evaluating limits.

    • Case 2: q(a) = 0 (zero case) requires simplification, potential cancellation of common factors.

    • Ensuring common factors in both numerator and denominator are removed before finding limits.

Limit Cases and Definitions

  • Common scenarios of limits:

    • Evaluate one-sided limits:

      • Left-sided and right-sided limits exist separately.

      • Result from both can be found from either piece of a piecewise defined function.

    • Analyze zero over non-zero limits, both signs matter:

      • Exploration of limits and behavior from different signs leads to outcome analysis.

    • Acknowledge behaviors across discontinuity points.

Example Evaluations of Limits

  • Work through examples step-by-step.

    • Example of evaluating f(x) as x approaches 3, using limit and function notation.

    • Identify forms like zero over zero situations to focus on common factors.

Special Cases in Limit Evaluation

  • Special examination for cases yielding 0/0 retains significance in calculus for completeness.

    • Final evaluations lead back to evaluating limits, whether defined or indeterminate.

  • Provide examples when one encounters limit scenarios with both positive and negative outcomes.

Visual Understanding of Functions and Limits

  • Graphs can give insights into behavior as x approaches specific points.

  • Always check for points of evaluations before plugging values.

  • Emphasis on graphical interpretation alongside algebraic evaluations.