All Flashcards
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.
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.
Formula for finding the y-coordinate of a hole at x = a.
Given r(x) = p(x)/q(x) with a hole at x=a, y = lim x->a [r(x) after canceling common factors].
Express the limit at a hole mathematically.
If r(x) has a hole at x = c, then , where L is the y-coordinate of the hole.
Factoring formula for .
General form of a rational function with a hole at x=a.
, where p(a) and q(a) are not simultaneously zero.
How to find the limit L at a hole x=c?
L = [Simplified r(x) after canceling common factors]
Write the general form of a factor in a polynomial.
, where 'a' is a root of the polynomial.
How to express the y-coordinate of a hole using the simplified function?
If the simplified function is s(x), then y-coordinate = s(a), where x=a is the x-coordinate of the hole.
Formula for factoring a cubic difference .
Formula for factoring a cubic sum .
Express the condition for a hole using limits.
exists, but is undefined.