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.
How to find holes in a rational function?
- Factor numerator and denominator. 2. Identify common factors. 3. Set common factor = 0 to find x-coordinate. 4. Cancel common factors. 5. Evaluate simplified function at x-coordinate to find y-coordinate.
Steps to find the limit at a hole.
- Identify the x-coordinate of the hole. 2. Cancel the common factor. 3. Evaluate the simplified function at the x-coordinate.
How to determine if a rational function has a hole at x=a?
- Check if (x-a) is a factor of both numerator and denominator. 2. If yes, then a hole exists at x=a.
How to simplify a rational function with a hole?
- Factor the numerator and the denominator. 2. Identify and cancel the common factors. 3. Write the simplified function.
How to verify the y-coordinate of a hole graphically?
- Graph the rational function. 2. Zoom in around the x-coordinate of the hole. 3. Observe the y-value the function approaches.
How to deal with multiple holes in a rational function?
- Factor completely. 2. Identify all common factors. 3. Find coordinates for each hole separately.
What is the first step when analyzing rational functions for holes?
Factor both the numerator and the denominator completely.
How do you find the x-value where a hole occurs?
Set the common factor (that appears in both numerator and denominator) equal to zero and solve for x.
What do you do after canceling the common factor to find the y-coordinate of the hole?
Substitute the x-value of the hole into the simplified rational function.
How to confirm a hole exists at a specific x-value?
Verify that the function is undefined at that x-value due to a common factor, but the limit exists as x approaches that value.