All Flashcards
What is a hole in a rational function?
A point where the function is undefined because a factor is present in both the numerator and denominator.
Define 'multiplicity' of a zero.
The number of times a zero appears as a factor in the factorization of a polynomial.
What is a removable discontinuity?
A point on a function's graph that can be 'removed' by redefining the function at that point (e.g., a hole).
What is a rational function?
A function that can be defined as a quotient of two polynomials.
How does a hole relate to limits?
The y-coordinate of the hole is the limit of the function as x approaches the x-coordinate of the hole.
Define limit of a function.
The value that a function approaches as the input (or argument) approaches some value.
What does it mean for a function to be undefined at a point?
The function does not have a defined value at that specific input value, often due to division by zero.
What is a factor of a polynomial?
An expression that divides evenly into the polynomial, leaving no remainder.
What does it mean to 'cancel' a common factor?
To divide both the numerator and denominator of a fraction by the same factor, simplifying the expression.
What is the x-coordinate of a hole?
The value 'c' where the common factor (x-c) equals zero.
What are the differences between a hole and a vertical asymptote?
Hole: Removable discontinuity, limit exists. | Vertical Asymptote: Non-removable discontinuity, limit is infinite or DNE.
Hole vs. Zero of a Function.
Hole: Factor cancels out. | Zero: Factor remains in numerator.
Removable vs. Non-Removable Discontinuity.
Removable: Can be 'fixed' by redefining the function. | Non-Removable: Cannot be 'fixed'; function approaches infinity or oscillates.
What are the differences between finding the x-coordinate of a hole and a vertical asymptote?
Hole: Set the common factor equal to zero. | Vertical Asymptote: Set the remaining factors in the denominator equal to zero.
Limit at a hole vs. Limit at a vertical asymptote.
Hole: Limit exists and is finite. | Vertical Asymptote: Limit is infinite or does not exist.
Rational function with a hole vs. Polynomial function.
Hole: Originally a rational function with common factors. | Polynomial: No division by a variable expression.
Hole vs. Point on a graph.
Hole: Function is undefined. | Point: Function has a defined value.
Graph of original function vs. Graph of simplified function (after removing hole).
Original: Hole is present. | Simplified: Hole is 'filled in'.
Factor in numerator only vs. Factor in both numerator and denominator.
Numerator Only: Zero of the function. | Both: Potential hole.
Hole vs. Jump Discontinuity
Hole: Limit exists. | Jump Discontinuity: Left and right limits exist but are not equal.
How does a hole appear on a graph?
As an open circle at a specific point, indicating the function is not defined there.
What does the graph of a rational function with a hole tell you about its limit at that point?
The y-value the function approaches as x approaches the hole's x-coordinate is the limit of the function at that point.
How can you visually differentiate between a hole and a vertical asymptote on a graph?
A hole is a single point discontinuity (open circle), while a vertical asymptote is a line the function approaches but never crosses, causing the function to tend towards infinity.
What does the smoothness of the graph around a hole indicate?
It indicates that the limit exists at that point, even though the function is undefined.
How to identify a hole from a graph?
Look for an open circle (or removable discontinuity) on the graph.
What does a hole on a graph imply about the function's domain?
The function's domain excludes the x-value of the hole.
What does it mean if a graph has a hole at x=a?
The function is not defined at x=a, but the limit as x approaches 'a' exists.
How does the graph of a simplified rational function relate to the original function with a hole?
The graph of the simplified function is identical to the original function, except it is defined at the x-value where the hole existed.
What does the presence of a hole suggest about the function's continuity?
The function is discontinuous at the x-value of the hole, but it's a removable discontinuity.
How to estimate the limit at a hole using a graph?
Trace the graph towards the x-value of the hole from both sides; the y-value the graph approaches is the estimated limit.