All Flashcards
Explain the concept of removable discontinuities.
A 'hole' in the graph that can often be 'removed' by simplifying the function (e.g., factoring). The limit exists, but the function is not defined at that point, or defined to a different value.
Explain the concept of jump discontinuities.
A sudden 'jump' in the graph where the left and right-hand limits are different. Common in piecewise functions.
Explain the concept of asymptote discontinuities.
The function approaches infinity (or negative infinity) as x approaches a certain value, creating a vertical asymptote.
Why are removable discontinuities important?
They often involve algebraic manipulation (factoring) to identify and remove them, a skill frequently tested.
Where do jump discontinuities typically occur?
Almost exclusively in piecewise functions, where the different pieces of the function do not connect.
What is the key characteristic of an asymptote discontinuity?
The function's values increase or decrease without bound as x approaches a specific value.
How do you identify a discontinuity in a graph?
Look for breaks, holes, jumps, or vertical asymptotes in the graph.
What is the relationship between limits and continuity?
For a function to be continuous at a point, the limit must exist at that point and be equal to the function's value.
Why is understanding discontinuities important in calculus?
Discontinuities affect the behavior of functions and are crucial for understanding limits, derivatives, and integrals.
How can factoring help identify discontinuities?
Factoring can reveal common factors that lead to removable discontinuities in rational functions.
What are the differences between removable and jump discontinuities?
Removable: Limit exists, point missing. | Jump: Left and right-hand limits differ.
What are the differences between removable and asymptote discontinuities?
Removable: Limit exists, point missing. | Asymptote: Function approaches infinity.
What are the differences between jump and asymptote discontinuities?
Jump: Finite difference between left and right limits. | Asymptote: Function approaches infinity.
Compare continuous and discontinuous functions.
Continuous: No breaks, limits exist and equal function values. | Discontinuous: Has breaks, holes, or jumps.
Compare left-hand and right-hand limits.
Left-hand: Limit as x approaches from the left. | Right-hand: Limit as x approaches from the right.
What are the differences between piecewise and rational functions?
Piecewise: Defined by different functions over different intervals. | Rational: Ratio of two polynomials.
Compare limits at removable discontinuities and limits at asymptote discontinuities.
Removable: Limit exists and is finite. | Asymptote: Limit is infinite or does not exist.
What is the difference between a function being undefined at a point and having a removable discontinuity?
Undefined: Function simply has no value at that point. | Removable: Limit exists even though the function is undefined or has a different value.
Compare the graphs of functions with jump discontinuities and functions with asymptote discontinuities.
Jump: Graph has a clear vertical jump. | Asymptote: Graph approaches a vertical line infinitely closely.
What is the difference between a 'hole' and a vertical asymptote?
Hole: A single point is missing. | Vertical Asymptote: The function approaches infinity.
What is a discontinuity?
A point where a function is not continuous; where you must lift your pencil to draw the graph.
Define a removable discontinuity.
A point where a function is not defined, but the limit exists. Often called a 'hole'.
Define a jump discontinuity.
A point where the left-hand limit does not equal the right-hand limit.
Define an asymptote discontinuity.
A point where the function approaches infinity (or negative infinity).
What is a piecewise function?
A function defined by multiple sub-functions, each applying to a certain interval of the main function's domain.
What is a rational function?
A function that can be expressed as the quotient of two polynomials.
What does it mean for a function to be continuous at a point?
The function is defined at that point, the limit exists at that point, and the limit equals the function's value at that point.
What is a left-hand limit?
The value a function approaches as the input approaches a given value from the left side.
What is a right-hand limit?
The value a function approaches as the input approaches a given value from the right side.
What is the domain of a function?
The set of all possible input values (x-values) for which the function is defined.