All Flashcards
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.
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.
How to identify a removable discontinuity in a rational function?
- Factor the numerator and denominator. 2. Cancel common factors. 3. The cancelled factor indicates the x-value of the discontinuity. 4. Evaluate the simplified function at that x-value to find the y-value of the 'hole'.
How to find the limit at a removable discontinuity?
- Factor and simplify the function. 2. Substitute the x-value of the discontinuity into the simplified function.
How to identify a jump discontinuity in a piecewise function?
- Identify the x-value where the function changes definition. 2. Find the left-hand limit at that x-value. 3. Find the right-hand limit at that x-value. 4. If the limits are different, there's a jump discontinuity.
How to determine if a piecewise function is continuous at a point?
- Check if the function is defined at the point. 2. Find the left-hand limit. 3. Find the right-hand limit. 4. If the limits exist and are equal to the function's value at that point, it's continuous.
How to identify an asymptote discontinuity?
- Look for values of x that make the denominator of a rational function equal to zero. 2. Check the limits as x approaches these values from both sides. 3. If the limits approach infinity (or negative infinity), there's an asymptote discontinuity.
How to redefine a function to remove a discontinuity?
- Identify the discontinuity. 2. Find the limit at that point. 3. Define a new piecewise function that equals the original function everywhere except at the discontinuity, where it equals the limit.
How do you find the value that makes a piecewise function continuous?
- Set the pieces of the function equal to each other at the point where they meet. 2. Solve for the unknown variable.
How to determine if a function is differentiable at a point of discontinuity?
A function cannot be differentiable at a point of discontinuity.
How to analyze the continuity of a function with absolute values?
- Rewrite the absolute value function as a piecewise function. 2. Analyze the continuity of the resulting piecewise function.
How to find the equation of a vertical asymptote?
- Set the denominator of the rational function equal to zero. 2. Solve for x. 3. The solution(s) are the equations of the vertical asymptotes.