Calculus: Limits and Function Behavior

These flashcards cover key terms and concepts related to limits, slopes, and function behavior as discussed in the calculus lecture.

Slope

The steepness or degree of inclination of a line, defined as the ratio of the vertical change to the horizontal change.

Tangent Line

A straight line that touches a curve at a single point and represents the slope of the curve at that point.

Instantaneous Slope

The slope of the tangent line at a particular point on a curve, representing the rate of change at that exact point.

Limit

A fundamental concept in calculus that describes the value a function approaches as the input approaches a specified point.

Vertical Asymptote

A line that a graph approaches but never touches or crosses, typically associated with a division by zero.

Open Circle

A notation used on graphs to indicate that a particular point is not included in the function.

Cubic Function

A polynomial function of degree three, which can exhibit different behaviors based on its coefficients and the values of x.

Parabolic Behavior

The shape of a graph that resembles a parabola, typically characterized by a quadratic function.

Piecewise Function

A function that is defined by different expressions or formulas depending on the input value.

Infinite Limit

A limit which approaches infinity (or negative infinity) as the variable approaches a given value.

Epsilon-Delta Definition of Limit

A formal mathematical definition of limits that describes how close the output of a function can get to a limit value when the input is sufficiently close to a target.

Oscillation

The behavior of a function that cycles between values, often without settling on a single limit.

Factorization

The process of breaking down an expression into simpler terms (factors) that multiply together to form the original expression.

Numerator and Denominator

The top and bottom parts of a fraction, respectively, which represent the quantity being considered and the quantity that divides it.

Error

A mathematical mistake or inconsistency, often resulting from incorrect assumptions or calculations.

Limit of Sin(1/x) as x approaches zero

This limit does not exist because the function oscilates about the point of interest

Limit of 1/x² as x approaches zero

This function never ceases to approach infinity, therefore this function has an infinite limit; technically the limit does not exist.

Limit of the absolute value of x/normal x as x approaches zero

As one x approaches zero it also approaches negative 1 while the other x approaches positive one, meaning the limit for this function does not exist.