Limits and Continuity

Understanding Limits and Discontinuities

Limits are fundamental in calculus, describing the behavior of functions as inputs approach specific values. Discontinuities occur when a function is not continuous at a point, often due to jumps, holes, or asymptotes.

• Limit Definition: The value that a function approaches as the input approaches a certain point.

• Types of Discontinuities:

  • Jump Discontinuity: The function "jumps" from one value to another.

  • Infinite Discontinuity: The function approaches infinity (vertical asymptote).

  • Removable Discontinuity: A hole in the graph where the function is not defined.

• Vertical Asymptote: A line where the function grows without bound as it approaches a specific x-value.

Example: For , the vertical asymptotes are at and where the denominator is zero.

Polynomial Functions

Degree, Leading Coefficient, and Graphs

Polynomial functions are expressions involving powers of x with real coefficients. The degree and leading coefficient determine the end behavior and shape of the graph.

• Degree: The highest power of x in the polynomial.

• Leading Coefficient: The coefficient of the term with the highest power.

• End Behavior: Determined by the degree and sign of the leading coefficient.

Example: For , the degree is 4 and the leading coefficient is -3.

Finding X-Intercepts

X-intercepts are points where the function crosses the x-axis, i.e., where .

• Set the function equal to zero and solve for x.

Example: For , set and solve for x.

Exponential and Logarithmic Equations

Solving Exponential Equations

Exponential equations involve variables in the exponent. They are solved using logarithms.

• General Form:

• Solution:

Example: Solve by taking the natural logarithm of both sides:

Compound Interest and Exponential Growth

Business calculus often applies exponential functions to model compound interest and growth.

• Compound Interest Formula (n times per year):

• Continuous Compounding Formula:

• Variables:

  • P: Principal (initial amount)

  • r: Annual interest rate (decimal)

  • t: Time in years

  • A: Amount after time t

Example: Find the future value of invested for $15 compounded monthly:

For continuous compounding:

Doubling and Tripling Time

To find the time required for an investment to double or triple, set or and solve for t.

• For continuous compounding:

Revenue and Profit Functions

Modeling Revenue and Profit

Revenue and profit functions are essential in business calculus for analyzing financial outcomes.

• Revenue Function: where is the price-demand function.

• Profit Function: where is the cost function.

• Domain: The set of x-values (units produced/sold) for which the functions are defined and meaningful.

Example: Given , ,

Finding Maximum Revenue and Profit

Maximum values occur at the vertex of a quadratic function .

• Vertex Formula:

Example: For , maximum revenue at

Graphical Analysis of Functions

Interpreting Graphs

Graphs provide visual insight into function behavior, including intercepts, asymptotes, and continuity.

• X-intercepts: Where the graph crosses the x-axis.

• Y-intercepts: Where the graph crosses the y-axis.

• Asymptotes: Lines the graph approaches but never touches.

• Vertex: The highest or lowest point on a parabola.

Example: For , the vertex is at .

Summary Table: Key Formulas and Concepts

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Graphical analysis and polynomial identification are common in business calculus for modeling and optimization.